Numerical Simulation of Fluid Flow and Heat transfer
in a Rotary Regenerator
M. Passandideh-Fard 1 ,M. Mousavi 2 and M. Ghazikhani3
1,3
Assistant Professors, Department of Mechanical Engineering, Ferdowsi University of Mashhad
Mashhad,Iran
2
Graduate Student, Department of Mechanical Engineering, Ferdowsi University of Mashhad
Mashhad,Iran
Email: mpfard@um.ac.ir
ABSTRACT
In this paper, a numerical simulation of fluid flow
and heat transfer in a rotary regenerator is presented.
The numerical method is based on the simultaneous
solution of the transient momentum and energy
equations using a finite volume scheme. To validate
the model, the numerical results are compared with
those of the analytical results for a simple geometry.
The effects of important processing parameters on the
rotary regenerator effectiveness
were also
investigated. The parameters considered were the
rotational speed, and the rotor length and diameter. In
the design of a regenerator, the mass flow rate of both
hot and cold streams and effectiveness are main
parameters. The developed model, therefore, is
suitable for the design and optimization of rotary
regenerators for industrial applications.
1.
alternatively (Fig. 1). First the hot fluid gives up its
heat to the regenerator; then the cold fluid flows
through the same passage picking up the stored heat.
The overall efficiency of sensible heat transfer for
this kind of regenerator can be as high as 85 percent,
while efficiency of plate-type heat exchangers is
between 45 and 65 percent [1].
INTRODUCTION
Heat recovery applications are becoming increasingly
more attractive as energy prices are raising
continuously. Heat exchangers are one of the
important equipment that can achieve this purpose.
There are many different types of heat exchangers
one of which is gas-to-gas used in some applications
such as furnaces, air conditioning systems, and power
plants. The main issue regarding the design of these
types of heat exchangers is the much lower heat
transfer coefficients of gases compared to those of
liquids. As a result, the required surface area of heat
transfers, and in turn, the sizes of these heat
exchangers will be much bigger compared to those of
gas-to-liquid or liquid-to-liquid heat exchangers.
Rotary regenerator is a solution for this problem. It
consists of a rotating matrix through which the hot
stream and cold stream flow periodically and
Fig. 1 A typical rotary regenerator and its extended
surface with large number of channels.
Rotary regenerator has a self cleaning property that
removes any fouling problem. Reversal of flows
causes this. Both compactness and high effectiveness
may increase the application type of this heat
exchanger type.
Studding on the rotary regenerator has a long history.
Sheiman and Reznikova were defining a model to
calculate heat transfer [2].They used integral laplace
transformation to solve differential equation.
Nahavandi and Weinstin [3] used closed methods in
mathematical model for solving the differential
equations. There are many attempts to obtain
empirical effectiveness relation [4-6], the ε-NTU0
method was used for the rotary regenerator design
and analysis. Organ [7] explains a procedure for
design regenerator. His formulation is the basis for an
analytical statement of regenerator operating at
optimum balance between pumping loss and penalty
of incomplete temperature recovery. Numerically
solutions are more attended to analysis this device
.Kovalevskii[8] solved the conjugate problem on
unsteady heat exchange between one-dimensional
flows and a two-dimensional matrix wall. The
optimum design parameters of such a regenerator and
rotational speeds of its rotor that which provide a
maximum heat efficiency of the regenerator at a
minimum aerodynamic drag in it have been
determined. Other authors [9,10] developed
Numerical solution. All of them were same in used
finite difference scheme for solving differential
equation. Effects of major parameters were
investigated in many papers. Shah and Skiepko[11]
and Drobnic and co workers [12] considered
Influence of leakage on the thermal performance.
Influence of rotational speed on effectiveness of
rotary heat exchanger was studied by Buyukalaca and
Yılmaz[13]. In the foregoing analysis, the influence
of longitudinal heat conduction was neglected.
Bahnke and Howard[14] and Romie[15] obtained
relationships,can be estimate the influence of
longitudinal heat conduction in the wall. Porowski
and Szczechowiak [16] investigated the effect of
longitudinal conduction in the matrix on effectiveness
of the rotary heat regenerator numerically .It is shown
large axial temperature gradient in the wall,
longitudinal heat conduction reduces the regenerator
effectiveness and the overall heat-transfer rate.
Sphaier and Worek [17] presented a 1D and
2Dsimulation of a rotary regenerator where they
concluded that a 2D formulation is needed for certain
design and operating parameters. In this paper, a
2D/axisymmetric model is presented for the simulation
of a rotary regenerator and the effects of important
processing parameters are investigated.
3. Change of angular momentum as the fluid enters
the rotating matrix is negligible.
4. Temperature and velocity of Inlet hot and cold
streams are constant during each period.
5. The matrix divided to two equal sections, so time
of hot and cold period is same.
The fluid flow and heat transfer through the channel is
axisymmetric (r and x coordinates) and the channel
radius is much smaller than its length. Therefore, the
governing equations are fluid flow equations in the
channel and energy equations in both fluid and solid
wall. These equations are written as:
u
1 
(ru r )  x  0
r r
x
u x
u x
u
 2u x
1 P
1  u x
 ur
 ux x  
 {
(r
)
}
t
r
x
 f x
r r
r
x 2
ur
u
u
1 P
 1 
 2u
 ur r  u x r  
 { (
(rur ))  2r }
t
r
x
 f r
r r r
x
 2T f
x
2
 2Tw
x 2
 f c pf T f
T f
T f
1  T f
(r
)
(
 ux
 ur
)
r r
r
kf
t
x
r

1  Tw
 c T
(r
) w w w
r r
r
k w t
where u is velocity; T temperature; P pressure;
ρ,υ,k and c are density, viscosity, thermal
conductivity coefficient and Specific heat capacity,
respectively. The subscript f refers to the fluid and w to
solid wall of the channel.
In equations which
mentioned above, ur , u x , p,T f and Tw are unknown
parameters. For better heat transfer between fluid and
solid wall, fluid flow is cosidered laminar in the
Insulated
Insulated
a)
2. NUMERICAL METHOD
The matrix of a rotary regenerator contains a large
number of channels as shown schematically in the inset
in Fig. 1. Flow velocity, temperature and other variables
are nearly the same for all channels. Therefore, to obtain
a numerical solution for this problem, we consider a
single channel of the matrix where hot and cold streams
flow periodically and in the opposite direction of each
other. The model is based on the following assumptions:
1. There is no fluid leakage from seals.
2. Fluid and solid physical properties are variable with
temperature throughout the periods.

Insulated
‫شرط‬
‫مرزی‬
Inlet B.C
‫عدم‬
Tin ,U in
‫انتقال‬
are known
‫حرارت‬
r
x
Axisymmetric
B.C
‫شرط‬
Insulated
Outlet
B.C
b)
‫مرزی‬
Insulated
‫تقار‬
‫ن‬
‫محور‬
‫ی‬
‫شرط‬
‫مرزی‬
‫عدم‬
Outlet
B.C
‫انتقال‬
‫شرط‬
‫حرارت‬
‫مرز‬
‫ی‬
‫خرو‬
‫جی‬
Insulated
Inlet B.C
Tin ,U in are
known
r
x
Axisymmetric B.C
Fig. 2 simplify aspect of a channel and its boundary
conditions which is used .a) cold period, b) hot
period.
channels of a rotary regenerator, generally.
Thereforelaminar flow is attended in numerical
method.The
equations
are
discretized
and
computationally solved using a control volume scheme.
SIMPLE algorithm is employed for solve pressurevelocity coupling problem. We used fully implicit
scheme for purpose transient calculations, because of its
robustness and unconditional stability [18].The
boundary conditions are periodic conditions at the two
ends of the channel and adiabatic at the outer surface of
the wall (Fig. 2). The input parameters are the entrance
velocity and temperature of the two hot and cold
streams.
The matrix covers the hot and cold flows equally;
therefore, the duration of hot and cold streams is equal
to each other (half the time of one revolution of the
matrix). At the beginning of each period, the initial
temperature is equal to its quantity at the end time of the
previous period. This procedure will be continuing
until the ultimate temperature of domain at each
period is very close to its quantity at the end of same
last periods (convergence criterion for unsteady
solution).
3. ANALYTICLAL METHOD
To validate the model, we consider fluid flow and heat
transfer in a simplified case for which analytical solution
can be obtained. The model results are then compared to
those of the analytical solution. Figure 3 displays this
simplified scenario.
A
,
4h
 f c f Dh
B
h
 wcw
Where h is convection heat transfer coefficient,
 , c and  are density, heat capacity and wall
thickness, respectively.To solve above diffential
equations , a limited method (the reason of calling is
the problem solved by two constant initial conditions)
is presented. In second step, we guessed the initial
temperature in form of power series. This method is
called close method and developed by Nahavandi and
Weinstin [3].
3.1 Simplified Limited Method
To simplify the problem in order to obtain an analytical
solution, the initial condition is homogenized by
assuming the temperature of the fluid domain to be Thi
during the heating period, and Tci during the cooling
period. The analytical problem stated above can be
solved using the method of Laplace transform.
Solving the equations yield:
x
 f ( x, t )   w ( x, t )  0
t
U
1
AB
 f ( x, t )   0e {e Bt* I 0 [2(
xt*) 2 ]  t  x
U
U
1
t*
AB
B  e B I 0 [2(
x ) 2 ]d }
0
U
 Ax
1
t*
AB
x
 w ( x, t )  B 0e U  e  B I 0 [2(
x ) 2 ]d t 
0
U
U
 Ax
U
In these equations,  f ,  w and  0 are defined:
Fig. 3 A single channel of the matrix of a rotary
regenerator (see the inset of Fig. 1).
Since the channel length is much bigger than its radius,
the heat transfer problem in radial direction can be
assumed lumped in the analytical problem.
uniform velocity of U is assumed throughout the
channel. The conduction heat transfer along the channel
in both liquid and solid wall is also neglected. The heat
transfer equations for fluid and wall are then simplified
as:
T f
U f
T f
 f ( x, t )  T f ( x, t )  T 
 w ( x, t )  Tw ( x, t )  T 
 0  Tci  Thi
Where h is convection heat transfer coefficient, T' is
equal to Tci in cold period and Thi hot period.
3.2 Close Method
Nahavandi and Weinstin [3] obtained an analytical
solution for the same problem without homogenizing the
initial condition. After a rather complex and lengthy
mathematics (called closed method), they solved the
differential equations and provided an estimate for the
effectiveness using ε-NTU0 method. Their result is:

 A(Tw  T f )
t
x
Tw
  B(Tw  T f )
t
A and B are defined:
T f ( , )  Thi  e    [Thi  f ( )]e I 0 (2  (   ) )d
0

Tw ( , )  Thi  e [Thi  f ( )]  e    [Thi  f ( )].

0
e


 
I 1 (2  (   ) )d
*
T f* ( *,*)  Tci  e  * *  [Tci  f * ( )]e  I 0 (2  * ( *  ) )d
0
T ( , )  Tci  e
*
w
 *
*
[Tci  f * ( *)]  e  * *  [Tci  f * ( )].
Tw,middle
360
0
*
I 1 (2  * ( *  ) )d
 * 
n 
n 
n 0
n 0
n
f ( )   a n  n , f * ( * )   a n * *
  
Hot period
Cold period
Tf,middle
300


Tw,middle
0
α , β and γ are written as:
 f c f U f Dh ,
,
4h
 
 f c f Dh
4h

 w c w
h
we considered n=3 in power series ,so need to solve
eight equation to obtain eight unknown parameters
( a0 , a1 , a2 , a3 , a0* , a1* , a2* , a3* ).for this porpose, we used
matlab software to solve the equations.
4. RESULTS AND DISCUSSION
Numerical model must be evaluated at first. We use
results of our analytical solutions to validate our
code. Then, numerical solution results are analyzing
and effect of some parameters on the rotary
regenerator effectiveness will be investigated. these
important parameters are: rotational speed, fluid inlet
speed and rotor length of rotary regenerator.
4.1 Model Validation
As mentioned before, our simplified method is
presented only for numerical model evaluation.
Figure 4 compares the analytical solution with that of
the numerical model for this simplified scenario. The
time variation of the temperature in the middle of the
channel for both solid wall and fluid is shown in the
figure during the hot and cold periods. It should be
noted that the numerical model provides a
temperature profile in each cross section of the
channel; therefore, in order to compare the numerical
and analytical results, we integrated the temperature
profile in r-direction to obtain a mean temperature in
each cross section.
5
10
15
20
Time(s)
25
30
Fig. 4 Comparison between the results of numerical
model (solid lines) and simplified analytical solution
(symbols) for a single channel (Fig. 2) during a hot
and cold period.
The result of their analytical solution for one complete
period of hot and cold flows is compared with that of the
numerical model in Fig. 5. The close comparisons
shown in both Fig. 4 and Fig. 5 between the simulations
and analytical
solutions validate the model and its underlying
assumptions.
340
Tf,middle
335
330
Temperature(K)


320
in these equation, (*) superscript refers to hot period
and non existence any superscript is related to cold
period. ζ and η are dimensionless length and
dimensionless time and defined:
x ,

 

 t  x



Tf,middle
340
Temperature(K)
e
325
Tw,middle
320
315
310
0
5
10
15
20
Time(s)
25
30
Fig. 5 Comparison between the results of numerical
model (solid lines) and analytical solution (symbols)
of ref.[3] for a single channel (Fig. 3) during a
continuous cold and hot period.
4.2 Results
A typical rotary regenerator with following geometrical
and operational parameters was considered:
- the length 0.45m,
- a single channel diameter 2mm,
- a single channel wall thickness 0.15mm,
- the entrance flow velocity in both hot and cold streams
4m/s, it is constant during the periods.
-inlet cold air temperature is 288.15 k and inlet hot air
temperature is 363.15 k.
-rotational speed is 2rpm; therefore, time of both hot
and cold period is 15s.
Since the hot and cold streams flow periodically and in
the opposite direction through the channels, the resulting
temperature at any point whether in the fluid or wall will
have a periodic shape. The results of simulations for the
average temperature of fluid and solid wall at the middle
point and at the two ends of the channel (left and right
sides) are displayed in Fig. 6. The rotational speed in
this case was 2 rpm. As observed, after around 270 s
from the start of the flows into the regenerator, we have
steady periodic shapes for all temperatures (except for
Tw,middle which shows an asymptotic behavior). This
means that after nearly nine periods (i.e. nine
revolutions of the regenerator matrix), the heat transfer
arrives at a steady condition.
370
channel with constant value, then quantity of it
decrease to zero, near the solid wall. This reduction
leads to increase the velocity in central places.
Finally, flow is fully developed in distance of 0.2m
from entrance. As we expected, U- velocity has
parabolic profile (relative to radius) in direction of
fluid flow. It is important that fluid is reaching to
steady state rapidly, between .25s and 15s velocity
value is changing insignificant. Both cold and hot
period is same in which mentioned above. The
difference between hot and cold period is in quantity
Tw,left
Temperature(K)
355
340
Tw,middle
(a)
Tf,left
325
310
Tf,middle
Tw,right
295
Tf,right
280
0
50
100
150
Time(s)
200
250
300
Fig.6- The average temperature variation of fluid and
solid wall at the middle point and at the two ends of a
single channel (left and right sides) of a regenerator
matrix. The rotational speed was 2 rpm.
the last revolution(revolution number 9) is the basis
our investigation; velocity, temperature and
effectiveness is investigated in this rotation.
The fluid flows are laminar in the channels. Velocity
vectors are shown in Fig. 7 at the various times. Floor
of chart is indicated place position and quantity of uvelocity is indicated in the normal direction of floor
plane. At hot and cold period flow has same shape
but in the opposite direction. Fluid is entering the
(b)
Fig.7- Velocity vectors in a channel at various times,
a) Cold period b) Hot period
of velocity. Density is variable with temperature
changing and so velocity values are different in hot
and cold period.
360
entry, heat exchange is preceded between fluid and
solid wall. There is two scenarios; Hot fluid is
heating solid wall and becomes cooler (cold period)
then, in the next period ,cold fluid is receiving
thermal energy that is storage in the solid wall (hot
period).It is observed ,temperature values in radius
direction is not constant and it is changed with going
far from solid wall. Therefore we have temperature
profile in each cross section and very small channel
radius is not leading a constant temperature.
Analytical methods were using the constant
temperature assumption (in radial direction), it is not
conforming to the fact, completely. Whatever time is
going forward, value of outlet temperature is neared
to its inlet quantity. Nearing inlet and outlet
temperature is not favorable for a heat exchanger,
thus a parameter is define to express capability of
regenerator:
340

Temperatures of fluid and solid wall are displayed in
Fig. 7. Floor of chart is indicated place position and
quantity of temperature is indicated in the normal
direction of floor plane. These temperatures are drawn
at various times ( t=0 indicates initial condition). Inlet
temperature is constant throughout a period. At initial
time, domain has the final temperature of last period.
hot fluid enters from left side in cold period and cold
fluid enters from right side in hot period. After fluid
T(k)
320
t=12.5 s
t=7.5 s
t=2.5 s
300
0
t=15 s
t=10 s
t=5 s
t=0.25 s
0s
t=
0.1
0.2
x(m)
0.3
0.002
0.001
)
r(m
0.4
(a)
360 t=2.5 s
t=
0s
t=7.5 s
T(k)
340 t=12.5 s
Tco  Tci
Thi  Tci
The overall heat-transfer performance of the
regenerator is most conveniently expressed as the
heat-transfer “effectiveness” ε, which compares the
actual heat-transfer rate to the thermodynamically
limited maximum possible heat-transfer rate. Tco is
outlet bulk temperature in the hot period and obtained
with temperature integrating in place and time. With
this definition, we can investigate effect of some
parameters on a rotary regenerator. They compare
with empirical relationship expressed by Kays and
London [6]:
*

1
1  e[  NTU ,0 (1C )] 


  (1 
)(
)
*
*1.93 1  C *e[  NTU , 0 (1C )]


9Cr


*
where heat capacity ratio, C ,and the number of
transfer units, Ntu;0, are defined as follows;
C
C*  min
C max
NTU , 0 
320
t=0.25 s
300
t=10 s
t=15 s
0.002
0.001
r(x
)
0.4
0.3
0.1
0.2x(m)
t=15 s
0
(b)
Fig.8- Temperature distribution in various time in a
channel, a) Cold period b) Hot period.

1 
1


C min 1 / hc Ac  1 / hh Ah 
Rotational speed; First, effectiveness variations with
different rotational speed are examined; the result is
shown in Fig.9. As seen in the figure, the effectiveness
is increasing with rotational speed of the regenerator.
Effectiveness is almost fixed after 1.5 rpm of
rotational speed, thus growth in rotational speed is
increasing effectiveness but not for ever. Therefore, it
needs to choose proper rotational speed (as seen 2rpm
was considered in previous study case).
Rotor length; Rotor length is another parameter that
affect on effectiveness. Its impact is shown in Fig.10.
Increase in length lead to effectiveness growth .in
fact, we investigate longitudinal conduction effect on
performance in some cases. Large axial temperature
gradient in the solid wall (longitudinal heat
conduction) reduces the regenerator effectiveness and
the overall heat transfer rate (It is known before, ideal
regenerator is that one has no axial conduction in the
wall), hence bigger length decrease axial conduction
and increase the effectiveness.
100
Effectiveness (%)
90
80
70
60
Keys&London
50
Numerical
40
30
20
0
0.5
1
1.5
2
2.5
Rotational Speed(rpm)
3.5 100
3
90
Effectiveness (%)
Fig.8- Variation of effectiveness with rotational
speed.
Rotational speed; First, effectiveness variations with
different rotational speed are examined; the result is
shown in Fig.9. As seen in the figure, the effectiveness
is increasing with rotational speed of the regenerator.
Effectiveness is almost fixed after 1.5 rpm of
rotational speed, thus growth in rotational speed is
increasing effectiveness but not for ever. Therefore, it
is needing to choose proper rotational speed (as seen
2rpm was considered in previous study case).
Inlet velocity; Inlet velocity (U) is a factor that has
strong effect on performance. It is investigated in
Fig.9.the effectiveness is decreasing with inlet
velocity increment. Lower speeds let better heat
transfer between fluid and solid wall .speeds are
indicated in the figure produce laminar flow (in 8m/s
Re= 2000). According to above, high inlet velocities
are not recommended, hence turbulence flow is not
attending in the rotary regenerator. Inlet velocity is
not independent variable. Mass flow rate and rotor
diameter assign the inlet velocity. In design, if mass
flow rate is considered constant, change in rotor size
result to U- velocity change.
Effectiveness (%)
100
90
Keys&London
80
Numerical
70
60
50
80
70
Keys&London
60
Numerical
50
40
30
0
0.5
1
1.5
Rotator Length(m)
2
Fig.10- Variation of effectiveness with rotor length.
5. CONCLUSIONS
In the study, a rotary regenerator was simulated by
solving a developed mathematical model. An
axisymmetric channel is considered for this aim. Two
analytical methods are expressed which include 1D
heat transfer. We employed them to validate our
numerical study and they were shown numerical
approach is approvable. U-velocity value is indicated
and observed a parabolic profile in direction of fluid
flow. We have a temperature profile in each cross
section too; it is showing that constant temperature
assumption is not exact reality. Effect of some
important parameter on the effectiveness of rotary
regenerator was investigated. As seen, increment in
rotational speed resulted increase in effectiveness
while inlet velocity increasing lead to decrease
performance. It is the reason that laminar condition is
considered in rotary regenerator. Finally rotor length
was investigated and concluded the longer length
present effectiveness increasing.
40
REFERENCES
30
0
1
2
3
4
5
6
Inlet Velocity(m/s)
7
8
Fig.9- Effect of inlet velocity on rotary regenerator
effectiveness.
9
2.5
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(a)
(b)
Fig.7- Temperature values in a channel at various
times, a) Cold period b) Hot period