Numerical Simulation of Fluid Flow and Heat transfer in a Rotary Regenerator

Transcription

1 Numerical Simulation o Fluid Flo and Heat transer in a Rotary Regenerator M. Mousavi 1, M. Passandideh-Fard and M. Ghazikhani 3 1 Graduate Student,, 3 Assistant Proessors Department o Mechanical Engineering, Ferdosi niversity o Mashhad, Mashhad, Iran mpard@um.ac.ir ABSTRACT In this paper, a numerical simulation o luid lo and heat transer in a rotary regenerator is presented. The numerical method is based on the simultaneous solution o the transient momentum and energy equations using a inite volume scheme. To validate the model, the numerical results are compared ith those o the analytical results or a simple geometry. The eects o important processing parameters on the rotary regenerator eectiveness ere also investigated. The parameters considered ere the rotational speed, and the rotor length and diameter. In the design o a regenerator, the mass lo rate o both hot and cold streams and the resulting heat eectiveness are main parameters. The developed model, thereore, is suitable or the design and optimization o rotary regenerators or industrial applications. same passage picking up the stored heat. The overall eiency o sensible heat transer or this kind o regenerator can be as high as 85 percent, hile eiency o plate-type heat echangers is beteen 45 and 65 percent [1]. 1. INTRODCTION Heat recovery applications are becoming increasingly more attractive as energy prices are raising continuously. Heat echangers are one o the important equipment that can achieve this purpose. There are many dierent types o heat echangers one o hich is gas-to-gas used in some applications such as urnaces, air conditioning systems, and poer plants. The main issue regarding the design o these types o heat echangers is the much loer heat transer coeients o gases compared to those o liquids. As a result, the required surace area o heat transers, and in turn, the sizes o these heat echangers ill be much bigger compared to those o gas-to-liquid or liquid-to-liquid heat echangers. Rotary regenerator is a solution or this problem. It consists o a rotating matri through hich the hot and cold streams lo periodically and alternatively (Fig. 1. First, the hot luid gives up its heat to the regenerator; then, the cold luid los through the Figure 1. A typical rotary regenerator and its etended surace ith a large number o channels. A rotary regenerator has a sel cleaning property that removes any ouling problem. This is caused by the periodic reversal los in the channels. This type o heat echanger is used idely because o its compactness and high eectiveness. Studying on the rotary regenerators has a long history. Sheiman and Reznikova [] introduced a model to calculate the involved heat transer. They used integral Laplace transormation to solve dierential equations. Nahavandi and Weinstin [3] used an elaborate mathematical model, hich they called Close method, to solve the dierential equations. There have been many attempts to obtain

2 an empirical eectiveness correlation [4-6]. The ε- NT method as used or the rotary regenerator design and analysis. Organ [7] eplained a procedure or designing a regenerator; his ormulation is the basis or a regenerator operating at an optimum balance beteen pumping loss and penalty o incomplete temperature recovery. Many numerical solutions are also available in the literature. Kovalevskii [8] solved the conjugate problem on unsteady heat echange beteen one-dimensional los and a to-dimensional matri all. He determined the optimum design parameters o such a regenerator and the rotational speed o its rotor or maimum heat eiency. Finite dierence scheme has been used by many researchers [9,1] to solve the involved dierential equation. Shah and Skiepko [11] and Drobnic et al. [1] considered the inluence o leakage on the thermal perormance. The eect o rotational speed on the eectiveness o rotary heat echanger as studied by Buyukalaca and Yılmaz [13]. They neglected the inluence o aial heat conduction. Bahnke and Hoard [14] and Romie [15] obtained relationships or estimating the inluence o aial heat conduction in the all. Poroski and Szczechoiak [16] investigated the eect o aial conduction in the matri on the eectiveness o the rotary regenerator. They shoed that a large aial temperature gradient in the all reduces the regenerator eectiveness and the overall heat-transer rate. Sphaier and Worek [17] presented a 1D and D simulations o a rotary regenerator here they concluded that a D ormulation is needed or certain design and operating parameters. In this paper, a D/aisymmetric model is presented or the simulation o a rotary regenerator and the eects o important processing parameters are investigated.. NMERICAL METHOD The matri o a rotary regenerator contains a large number o channels as shon schematically in the inset in Fig. 1. Flo veloty, temperature and other variables are nearly the same or all channels. Thereore, to obtain a numerical solution or this problem, e consider a single channel o the matri here hot and cold streams lo periodically and in the opposite direction o each other. The model is based on the olloing assumptions: Fluid and solid-all thermophysical properties may vary ith temperature. The change o the luid angular momentum as it enters the rotating matri is negligible. The inlet temperature and veloty o hot and cold streams remain unchanged in each period. The matri is divided to to equal sections, so the time duration o hot and cold periods is same. The luid lo and heat transer through the channel is aisymmetric (r and coordinates and the channel radius is much smaller than its length. Thereore, the governing equations are luid lo equations in the channel and energy equations in both luid and solid all. These equations are ritten as: 1 u ( rur + = (1 r u u u 1 P 1 u u + ur + u = + υ{ ( r + } ( ρ r ur ur ur 1 P 1 ur + ur + u = + υ{ ( ( rur + } (3 ρ r T 1 T ρ c + ( r = r k T 1 T ρc + ( r = r k p T ( + u T T T + ur here u is the veloty, T temperature, P pressure and ρ, υ, k and c are density, viscosity, thermal conductivity coeient and speic heat capaty, respectively. The subscript reers to the luid and to the solid-all o the channel. In the above equations, u r, u, p, T and T are unknon parameters. The channels are small in radius, thereore; the luid lo may be ell considered to be laminar. a b Inlet B.C T in, in are knon r r Outlet B.C Aisymmetric B.C Aisymmetric B.C Outlet B.C Figure. A schematic o the computational domain and boundary conditions or a cold period, b hot period. The equations are discretized and computationally solved using a control volume scheme. The SIMPLE (4 (5 Inlet B.C T in, are in knon

3 algorithm is employed to solve the pressure-veloty coupling problem. A ully implit scheme is used to treat the transient calculations, because o its robustness and unconditional stability [18]. The boundary conditions are periodic conditions at the to ends o the channel and adiabatic at the outer surace o the all (Fig.. The input parameters are the entrance veloty and temperature o the to hot and cold streams. The matri covers the hot and cold los equally; thereore, the duration o hot and cold streams is equal to each other (one hal o the time duration o one revolution o the matri. At the beginning o each period, the initial temperature is equal to its quantity at the end time o the previous period. This procedure continues until the temperatures at dierent points o the channel at each period are very close to the corresponding temperatures at the end o the previous period (convergence criterion or unsteady solution. 3. ANALYTICAL METHOD To validate the numerical model, e consider the luid lo and heat transer in a simpliied case or hich an analytical solution can be obtained. The model results are then compared to those o the analytical solution. Figure 3 displays this simpliied scenario. 4h A =, B = ρ c D h h ρ c δ here h is the convection heat transer coeient, and ρ, c andδ are density, heat capaty and all thickness, respectively. To solve the above coupled partial dierential equations, to methods are presented; one is based on a simpliied case here the initial temperature is assumed constant and the other one is a more elaborate method based on a guess or the initial temperature in the orm o poer series, irst introduced by Nahavandi and Weinstin [3]. 3.1 Simpliied method To simpliy the problem in order to obtain an analytical solution, the initial condition is homogenized by assuming the temperature o the luid domain to be T hi during the heating period, andt during the cooling period. The analytical problem stated above can be solved using the method o Laplace transorm. Solving the equations yields: θ (, = θ (, = θ (, = θ e B A t 1 Bt AB { e I[( ] + 1 Bτ AB e I[( τ ] dτ} t < t (8 (9 θ (, = Bθ e A t 1 Bτ e ] AB I [( τ dτ t (1 In these equations, θ, θ and θ are deined: Figure 3. A single channel o the matri o a rotary regenerator (see the inset o Fig. 1. Since the channel length is much bigger than its radius, the heat transer problem in the radial direction can be assumed lumped in the analytical problem. A uniorm veloty o is assumed throughout the channel. The conduction heat transer along the channel in both the luid and solid-all is also neglected. The heat transer equations or the luid and all are then simpliied as: T T + = B T = A T T ( T ( T here A and B are deined as: (6 (7 θ (, = T (, T (11 θ (, = T (, T (1 θ = T (13 T hi here h is the convection heat transer coeient, and T' is equal to T in the cold period and T hi in the hot period. 3. Close method Nahavandi and Weinstin [3] presented an analytical solution or the same problem ithout homogenizing the initial condition. Ater a rather comple and lengthy mathematics (hich they called Close method, they solved the dierential equations and provided an

4 estimate or the eectiveness using ε-nt method. Their result reads: ξ ξ η ε T ( ξ, η = Thi e [ Thi ( ε ] e I( η( ξ ε dε η ξ η T ( ξ, η = Thi e [ Thi ( ξ ] + e [ T e ε T ( ξ, η = T T ( ξ, η = T ξ ξ η I1( η( ξ ε dε ξ ε e ξ η ξ hi ( ε ]. ε [ T ( ε ] e I ( η ( ξ ε dε e η [ T ( ε ]. e. [ T ( ξ] + e ε ξ η η I1( η ( ξ ε dε ξ ε. (14 (15 (16 (17 n n ξ = ( = a n ξ (18 n= n n ( ξ = a n ξ n= = (19 here parameters ith asterisk ( reer to the hot period and those ithout it to the cold period. ζ and η are the nondimensionalized length and time deined as: ξ =, γ β η = t β αβ γ here α, β and γ are ritten as: ρ c δ α =, ρ c Dh β =, ρ c Dh γ = h 4h 4h We considered n=3 in the above-mentioned poer series; thereore, e needed to solve eight equations to obtain eight unknon parameters o a, a, 1 a, a, 3 a, a 1, a, and a 3. The MATLAB sotare as used or this purpose. 4. RESLTS AND DISCSSION In this section, irst a comparison beteen the results o the numerical model ith those o the analytics or the simple case o a single channel is presented. Having validated the model, then the simulation results or the complete luid lo and heat transer or a rotary regenerator ill be presented. The eect o important parameters on the eectiveness o a rotary regenerator ill also be investigated; they include: rotational speed, luid inlet veloty, and the rotor length o the regenerator. 4.1 Model validation As mentioned beore, the simpliied analytical method is presented only or numerical model evaluation. Figure 4 compares the analytical solution ith that o the numerical model or this simpliied scenario. The time variation o the temperature in the middle o the channel or both solid all and luid is shon in the igure during the hot and cold periods. It should be noted that the numerical model provides a temperature proile in each cross section o the channel; thereore, in order to compare the numerical and analytical results, e integrated the temperature proile in r-direction to obtain a mean temperature in each cross section. Temperature(K T,middle T,middle T,middle T,middle Time(s Hot period Figure 4. Comparison beteen the results o numerical model (solid lines and simpliied analytical solution (symbols or a single channel (Fig. during a hot and cold period. Figure 4 shos a very close agreement beteen analytical results and simulations or the simpliied case o periodic heat transer in a single channel. Net, e compared the more elaborate analytical results o Close method ith that o the numerical model. The result o the analytical solution or one complete period o hot and cold los is compared ith that o the numerical model in Fig. 5. The close comparisons shon in both Fig. 4 and Fig. 5 beteen the simulations and analytical solutions validate the model and its underlying assumptions. Cold period

5 34 37 T,let 335 T,middle 355 Temperature(K T,middle Temperature(K T,middle T,middle T,let T,right T,right Time(s Time(s Figure 5. Comparison beteen the results o the numerical model (solid lines and analytical solution (symbols o Close method [3] or a single channel (Fig. during a continuous cold and hot period. 4. Numerical results A typical rotary regenerator ith olloing geometrical and operational parameters as considered: the length.45m, a single channel diameter mm, a single channel all thickness.15mm, the entrance lo veloty in both hot and cold streams 4m/s (constant during the periods inlet cold air temperature 15 o C and inlet hot air temperature 9 o C, rotational speed rpm (i.e., the time duration o each period, hot or cold, is 15s. Since the hot and cold streams lo periodically and in the opposite direction through the channels, the resulting temperature at any point hether in the luid or all ill has a periodic shape. The results o simulations or the average temperature o luid and solid all at the middle point and at the to ends o the channel (let and right sides are displayed in Fig. 6. The rotational speed in this case as rpm. As observed, ater around 7 s rom the start o the los into the regenerator, e have steady periodic shapes or all temperatures (ecept or T,middle hich shos an asymptotic behavior. This means that ater nearly nine periods (i.e. nine revolutions o the regenerator matri, the heat transer arrives at a steady condition. Figure 6. The average temperature variation o luid and solid all at the middle point and at the to ends o a single channel (let and right sides o a regenerator matri. The rotational speed as rpm. The last revolution (revolution number 9 is the basis o our investigation to obtain veloty, temperature and eectiveness. Veloty vectors and temperature distribution in a channel is shon in Fig. 7 in successive times. The luid los are laminar in the channels. During hot and cold periods, the lo has same shape but in the opposite direction. Fluid is entering the channel ith a constant veloty and temperature. Due to no-slip condition at the solid all, the veloty proile gets a parabolic shape in the channel. The luid lo is ully developed in a distance o about.m rom the entrance. It is interesting to note that luid reaches a steady state condition very rapidly; rom t=.5s to t=15s the veloty magnitude does not change signiicantly. Temperature distributions o the luid are also displayed in Fig. 7 at the bottom o the corresponding veloty proiles. The inlet temperature is assumed constant. At the beginning o a period, the luid domain has the inal temperature obtained rom the last period. The hot luid enters rom the let side in the cold period, and the cold luid enters rom the right side in the hot period. Ater luid entry, heat echange occurs beteen the luid and solid all. Hot luid heats the solid all during the cold period. Then during the hot period, the heated all transers its stored energy to the cold luid. As seen in Fig. 7, in contrast to the veloty distributions, the temperature values vary in both aial and radial directions. Although the channel radius is too small compared to its length, the temperature variation in radial direction is signiicant. Thereore, the assumption o constant temperature proile in radial direction in analytical solutions is only an approimation.

6 u T (a (b Figure 7. Flo veloty vectors in aial direction and temperature distribution (lines ithout arro o the luid at dierent locations in a channel at successive times during a cold period and b hot period. The overall heat-transer perormance o a regenerator is most conveniently epressed as the heat-transer eectiveness ε, hich compares the actual heat-transer rate to the thermodynamically limited maimum possible heat-transer rate. This value is deined as: Tco T ε = ( T T hi here T co is the outlet bulk temperature in the hot period and is obtained by integrating the temperature ith regard to both time and location. With this

7 deinition, e can investigate the eect o some parameters on a rotary regenerator. The results can then be compared ith those o the empirical correlation given by Kays and London [6]: [ NT, (1 C ] 1 1 e ε = (1 ( [, (1 ] (1 NT C C 1 C e r here heat capaty ratio, C, and the number o transer units, Ntu, are deined as ollos; Cmin C = ( C ma 1 1 NT, = (3 Cmin 1/ hc Ac + 1/ hh Ah Rotational speed. First, the eect o rotational speed on the regenerator eectiveness is eamined; the result is shon in Fig.8. As seen in the igure, the eectiveness is increasing ith the rotational speed o the regenerator. The eectiveness approaches a constant value ater a speed o about 1.5 rpm, thus the eect o rotational speed on the eectiveness is bounded. A speed o rpm seems to be an optimum value as considered in the previous case study. Eectiveness (% Keys&London Numerical Rotational Speed(rpm Figure 8. Variation o the eectiveness ith rotational speed rom numerical model and empirical correlations. Inlet veloty. The lo inlet veloty ( is a actor that has strong eect on the regenerator perormance. The results are seen in Fig.9. The eectiveness is decreased as the inlet veloty is increased. This as epected because a loer luid veloty results in a better heat transer beteen luid and solid all. It should be mentioned that the veloties eamined here ere in the range that no turbulence as triggered (or the maimum veloty o 8m/s the Reynolds number as. As a result, a high inlet veloty is not recommended or a rotary regenerator. The inlet veloty is not an independent variable; its value is assigned by the mass lo rate and rotor diameter. In the design o a regenerator, ater the mass lo rate is set, the rotor size is selected such that it results in a proper - veloty. Eectiveness (% Keys&London Numerical Inlet Veloty(m/s Figure 9. Eect o inlet veloty on the eectiveness o a rotary regenerator rom numerical model and empirical correlations. Rotor length. The rotor length is another parameter that aects the eectiveness. Its impact is shon in Fig.1. Increasing the rotor length leads to an increase in the eectiveness. Large aial temperature gradient in the solid all (aial heat conduction reduces the regenerator eectiveness and the overall heat transer rate. Theoretically, it is knon that an ideal regenerator is the one that has no aial conduction in the all. Hence a longer rotor decreases the eect o aial conduction and, thereore, increases the eectiveness. Eectiveness (% Keys&London Numerical Rotator Length(m Figure 1. Variation o the eectiveness ith rotor length rom numerical model and empirical correlations. 5. CONCLSIONS In this study, a mathematical model as developed to simulate the luid lo and heat transer in a rotary regenerator. An aisymmetric channel as considered or this purpose. To analytical methods

8 based on 1D model o heat transer ere presented. They ere used to validate the numerical model or a simple geometry. A very close agreement as obtained beteen numerical results and those o the analytical models. The veloty distribution and the luid temperature proile ere obtained rom the model. While the luid lo in the channels became ully developed in a short distance rom the entry, the luid temperature varied in both aial and radial directions throughout the channel. Eect o some important parameter on the eectiveness o a rotary regenerator as also investigated. These results ere also compared ith those o the empirical correlations here a good agreement as observed. Increasing the rotational speed resulted in an increase in the eectiveness hile increasing the inlet veloty led to a decrease o the regenerator perormance. This is the reason that laminar lo conditions are preerred in rotary regenerators. The eect o rotor length as also investigated; increasing the rotor length increased the eectiveness. REFERENCES [1] T. Yılmaz and O. Buyukalaca. Design o Regenerative Heat Echangers. Heat Transer Engineering, 4(4:3 38, 3 [] V. A. Sheyman and G. E. Reznikova. A Method to Determine the Packing and Gas Temperatures in a Rotating Regenerator ith Disperse Packing. Inzhenerno-Fizicheskii Zhurnal, Vol. 13, No. 4, pp , 1967 [3] A. H.Nahavandi, and A. S. Weinstein. A solution to theperiodic lo regenerative heat echanger problem. Appl. S.Res, 1, ,1961 [4] J. E.Coppage and A. L. London. The periodiclo regenerator-a summary o design theory. Trans. ASME, 75, ,1953 [5] F. W.Schmidt and A. J. Willmott. Thermal Energy Storage and Regeneration. McGra-Hill, Ne York,1981 [6] A. L.London and K. M. Kays. Compact Heat Echangers. third edition, McGra-Hill, NeYork, 1984 [7] A.J. Organ. Analysis o the gas turbine rotary regenerator. Proc Instn Mech Engrs,Vol 11 Part D,1997 [8] V. P. Kovalevskii. Simulation o Heat and Aerodynamic Process in Regenerators o continuous and Periodic Operation. II. Investigation o the Parameters o a Gas-Turbine Plant Regenerator Operating in Dynamic and Quasi-Stationary Regimes. Journal o Engineering Physics and Thermophysics, Vol. 77, No. 6, 4 [9] J. Frauhammer,H. Klein, G. Eienberger,. Noak. Solving moving boundary problems ith an adaptive moving grid method rotary heat echangers ith condensation and evaporation. Konrad-Zuse-Zentrum r Inormationstechnik Berlin, 1996 [1] T. Skiepko and R.K. Shah. A comparison o rotary regenerator theory and eperimental results or an air preheater or a thermal poer plant. Eperimental Thermal and Fluid Sence, , 4 [11] T. Skiepko and R.K. Shah. Inluence o leakage distribution on the thermal perormance o a rotary regenerator. Applied Thermal Engineering, , 1999 [1] B.Drobnic, J.Oman, M.Tuma. A numerical model or the analyses o heat transer and leakages in a rotary air preheater. International Journal o Heat and Mass Transer, , 6 [13] O. Buyukalaca and T. Yılmaz. Inluence o rotational speed on eectiveness o rotary-type heat echanger. Heat and Mass Transer, , [14] G. D. Bahnke and C. P. Hoard. The eect o longitudinal heat conduction on periodic lo heat echanger perormance. Trans. ASME, J. Eng. Poer, 15-1,1964 [15] F. E.Romie. Treatment o transverse and longitudinal heat conduction in regenerators. Trans. ASME, J. Heat Transer, 113, 47-49, 1991 [16] M. Poroski and E. Szczechoiak. Inluence o longitudinal conduction in the matri on eectiveness o rotary heat regenerator used in air-conditioning. Heat Mass Transer,43:1185 1, 7 [17] L. A. Sphaier and W. M. Worek, Comparisons beteen -Dand 1-D ormulations o heat and mass transer in rotary regenerators, Numerical Heat Transer, Part B, 49: 3 37, 6 [18] H. K. Versteeg and W. Malalasekera. An introduction to computational luid dynamics The inite volume method. Longman Sentiic & Technical, 1995