Numerical Simulation of Fluid Flow and Heat transferin a Rotary Regenerator

M. Passandideh-Fard 1 ,M. Mousavi 2 and M. Ghazikhani3

1,3Assistant Professors, Department of Mechanical Engineering, Ferdowsi University of MashhadMashhad,Iran

2 Graduate Student, Department of Mechanical Engineering, Ferdowsi University of MashhadMashhad,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. 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 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].

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.

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.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:

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, and

are unknown parameters. For better heat transfer between fluid and solid wall, fluid flow is cosidered laminar in the

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 pressure-velocity coupling problem. We used fully implicit scheme for purpose transient calculations, because of its robustness and unconditional

Inlet B.Care known

شرط مرزی خروجی

شرط مرزی تقار

ن محوری

شرط مرزی عدم انتقال حرارت

شرط مرزی عدم انتقال حرارت

Insulated

Axisymmetric B.C

Inlet B.Care known Outlet

B.C

Insulated Insulated

Insulated

Axisymmetric B.C

Outlet B.C

Insulated Insulated

a)

b)

xr

xr

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.

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:

A and B are defined:

,

Where h is convection heat transfer coefficient, 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 MethodTo 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 during the heating period, and during the cooling period. The analytical problem stated above can be solved using the method of Laplace transform. Solving the equations yield:

In these equations, and are defined:

Where h is convection heat transfer coefficient, T' is equal to in cold period and hot period.

3.2 Close MethodNahavandi 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:

,

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:

,

α , β and γ are written as:

, ,

we considered n=3 in power series ,so need to solve eight equation to obtain eight unknown parameters (

, , , , , , , ).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 ValidationAs 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.

Time(s)

Tem

pera

ture

(K)

0 5 10 15 20 25 30

300

320

340

360

Tf,middle

Tw,middle

Tw,middle

Tf,middle

Cold period

Hot period

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 analyticalsolutions validate the model and its underlying assumptions.

Time(s)

Tem

pera

ture

(K)

0 5 10 15 20 25 30310

315

320

325

330

335

340

Tw,middle

Tf,middle

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 ResultsA 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 middlepoint 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.

280

295

310

325

340

355

370

0 50 100 150 200 250 300Time(s)

Tf,left

Tw,left

Tf,right

Tw,rightTf,middle

Tw,middle

Fig.6- The average temperature variation of fluid and solid wall at the middle point

and at the two ends of asingle 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 u-velocity 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 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

(a)

(b)

Fig.7- Velocity vectors in a channel at various times, a) Cold period b) Hot

periodof velocity. Density is variable with temperature changing and so velocity values are different in hot and cold period.

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

x(m)

00.1

0.20.3

0.4 r(m)0.001

0.002

T(k)

300

320

340

360

t=0.25 s

t=0 s

t=15 st=10 s

t=2.5 s t=5 s

t=12.5 st=7.5 s

(a)

x(m)

00.1

0.20.3

0.4

r(x)0.001

0.002

T(k)

300

320

340

360

t=15 s

t=15 s

t=2.5 s

t=7.5 st=12.5 s

t=0 s

t=10 s

t=0.25 s

(b)

Fig.8- Temperature distribution in various time in a channel, a) Cold period b) Hot

period.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:

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. 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]:

where heat capacity ratio, ,and the number of transfer units, Ntu;0, are defined as follows;

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).

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5 3 3.5Rotational Speed(rpm)

Keys&LondonNumerical

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.

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8 9Inlet Velocity(m/s)

Keys&London

Numerical

Fig.9- Effect of inlet velocity on rotary regenerator effectiveness.

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.

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5Rotator Length(m)

Effe

ctiv

enes

s (%

)

Keys&London

Numerical

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.

REFERENCES

[1] T. Yılmaz and O. Buyukalaca . Design of Regenerative Heat Exchangers. Heat Transfer Engineering, 24(4):32–38, 2003[2] V. A. Sheyman and G. E. Reznikova. A Method to Determine the Packing and Gas Temperatures in a Rotating Regenerator with Disperse Packing. Inzhenerno-Fizicheskii Zhurnal, Vol. 13, No. 4, pp. 455-462, 1967[3] A. H.Nahavandi, and A. S. Weinstein. A solution to theperiodic flow regenerative heat exchanger problem. Appl. Sci.Res, 10, 335-348 ,1961[4] J. E.Coppage and A. L. London. The periodic-flow regenerator-A summary of design theory.Trans. ASME, 75, 779-787 ,1953 [5] F. W .Schmidt and A. J. Willmott. Thermal Energy Storage and Regeneration. McGraw-Hill,New York,1981[6] A. L .London and K. M. Kays. Compact Heat Exchangers. third edition, McGraw-Hill, NewYork, 1984[7] A .J. Organ. Analysis of the gas turbine rotary regenerator. Proc Instn Mech Engrs ,Vol 211 Part D,1997[8] V. P. Kovalevskii. Simulation of Heat and Aerodynamic Process in Regenerators of continuous and Periodic Operation. II. Investigation of the Parameters of a Gas-Turbine Plant Regenerator Operating in Dynamic and Quasi-Stationary Regimes. Journal of Engineering Physics and Thermophysics, Vol. 77, No. 6, 2004[9] J. Frauhammer ,H. Klein, G. Eiffenberger, U. Nowak . Solving moving boundary problems with an adaptive moving grid method rotary heat exchangers with condensation and evaporation. Konrad-Zuse-Zentrum fr Informationstechnik Berlin, 1996 [10] T. Skiepko and R.K. Shah. A comparison of rotary regenerator theory and experimental results for an air preheater for a thermal power plant. Experimental Thermal and Fluid Science, 28 257–264, 2004[11] T. Skiepko and R.K. Shah. Influence of leakage distribution on the thermal performance of a rotary regenerator. Applied Thermal Engineering, 19 685-705, 1999[12] B.Drobnic, J.Oman, M.Tuma. A numerical model for the analyses of heat transfer and leakages in a rotary air preheater. International Journal of Heat and Mass Transfer ,49 5001–5009, 2006

[13] O. Buyukalaca and T. Yılmaz . Influence of rotational speed on effectiveness of rotary-type heat exchanger. Heat and Mass Transfer, 38 441-447, 2002[14] G. D. Bahnke and C. P. Howard. The effect of longitudinal heat conduction on periodic flow heatexchanger performance. Trans. ASME, J. Eng. Power, 105-120,1964[15] F. E.Romie. Treatment of transverse and longitudinal heat conduction in regenerators. Trans.ASME, J. Heat Transfer, 113, 247-249, 1991[16] M. Porowski and E. Szczechowiak. Influence of longitudinal conduction in the matrix on effectiveness of rotary heat regenerator used in air-conditioning. Heat Mass Transfer,43:1185–1200, 2007[17] L. A. Sphaier and W. M. Worek, Comparisons between 2-Dand 1-D formulations of heat and mass transfer in rotary regenerators, Numerical Heat Transfer, Part B, 49: 223–237, 2006[18] H. K. Versteeg and W. Malalasekera. An introduction to computational fluid dynamics The finite volume method. Longman Scientific & Technical, 1995

(a)

(b)

Fig.7- Temperature values in a channel at various times, a) Cold period b) Hot

period