Description of a One-Dimensional Numerical Model of an Active Magnetic Regenerator Refrigerator

Transcription

1 This is a 1D model o an active magnetic regenerative rerigerator (AMRR) that was developed in MATLAB. The model uses cycle inputs such as the luid mass low and magnetic ield proiles, luid and regenerator material properties, and regenerator geometry properties to generate the cyclical steady state temperature proile o the luid and regenerator. Using the temperature proiles, the cooling load produced by the system and work input to the system are calculated. The development o the model is discussed in Progress Report #1. This model only considers the magnetic regenerator and does not model heat exchangers or other external hardware. The model starts rom an initial temperature proile or the regenerator and luid and steps orward in time using implicit time steps until cyclical steady state is achieved. The user must deine the number o time steps in each cycle and the number o nodes in the axial direction in the regenerator. Modeling parameters related to operating conditions, material properties, and geometry are determined by user-deined unctions. New unctions or each may be written to model systems that are not ully deined by the unctions provided here.

2 Description o a One-Dimensional Numerical Model o an Active Magnetic Regenerator Rerigerator 1. Governing Equations Figure 1 shows a schematic o an active magnetic regenerator modeled as a one dimensional (1D) system. The equipment that is external to the bed (e.g., the pumps, heat exchangers, and permanent magnets), are not explicitly modeled; however, their impact on the cycle is elt through an imposed time variation o the mass low rate ( mt () ) and the variation o the magnetic ield in time and space ( (, ) µ H x t o ). The time variation o these quantities is related to the luid-mechanical-magnetic processes associated with the cycle implementation. The interace between these imposed boundary conditions and the characteristics o these auxiliary pieces o equipment can be handled by system-level models that can interact with this component-level model. x heat transer luid ( ), ρ, ( ), µ ( ) c T k T T T (x,t) hot end, mt () luid enters at T H regenerator bed Ac, L, a s, d h, k e, ε T Curie (x) Nu(Re, Pr ) (Re ) µ o H( x, t) magnetic regenerator material: cold end, luid enters at T C T r (x,t) s k T T T H c T T H T ( ), r (, µ ), (,, µ ), ρ ( ) r r Curie o r Curie o r Curie µ oh T Figure 1. Conceptual drawing o a 1D AMR model showing the important parameters A positive luid mass low rate indicates that low is in the positive x direction, as indicated in Figure 1 and thereore enters the hot end o the regenerator bed; when it is negative it enters at the cold end. The luid is assumed to be incompressible and thereore there can be no time variation in the mass o luid that 1

3 is entrained in the bed. Continuity indicates that the mass low rate must be spatially uniorm within the bed so that the speciication o the mass low rate at the boundaries is suicient to determine the mass low rate throughout the bed. The low entering the bed is assumed to have the temperature o the adjacent thermal reservoir, T H or T C depending on whether the low rate is positive or negative, respectively. The required luid properties include the density (ρ ), speciic heat capacity (c ), viscosity (µ ), and thermal conductivity (k ). The speciic heat capacity, viscosity, and thermal conductivity are assumed to be some unction o temperature but not pressure. The density o the luid is assumed to be unaected by either temperature or pressure. The luid lows within a regenerator matrix composed o a magnetic material. The magnetic material may be layered; this layering may be represented simply as a spatial variation in the Curie temperature (T Curie (x)) or, in more detail, as material properties that depend on the axial location within the bed. The partial derivative o the speciic entropy o the material with respect to applied ield at constant temperature is a unction o the temperature o the material and o the applied magnetic ield s µ H r ( ( T, µ H) o T o ). The speciic heat capacity o the material at constant applied ield o the material is assumed to be a unction o the material s temperature and applied ield and the conductivity is assumed to be a unction o temperature ( c ( T, H) and k ( T) and thereore has a constant density (ρ r ). µ µ ). The material is assumed to be incompressible o H o r The geometry o the matrix must consist o many small passages that place the luid in intimate thermal contact with the regenerator material. Regenerator geometries ranging rom packed beds o spheres to screens to perorated plates may all be considered by this model by adjusting the thermal-luid

4 correlations and the geometric parameters. In order to maintain this lexibility, the regenerator geometry is characterized by a hydraulic diameter (d h ), porosity (ε), and speciic surace area (a s ). The Nusselt number o the matrix is assumed to be a unction o the local Reynolds number and Prandtl number o the luid (Nu(Re, Pr )). The riction actor is assumed to be a unction o the local Reynolds number ((Re )) and geometry. This riction correlation is suicient or a steady or slowly modulating lows; however more sophisticated correlations requiring additional parameters may be required to characterize the oscillatory nature o the low. The matrix is characterized by an eective static thermal static conductivity (k e ) that relates the actual, axial conduction heat transer in the absence o luid low to the heat transer through a comparable solid piece o material. Axial dispersion due to the eddy mixing o the luid during luid low is treated as an augmented thermal conductivity in the luid (k disp ). The values o these parameters depend on the particular geometry, materials, and low conditions that are simulated. The overall size o the regenerator is speciied according to its length (L) and total cross-sectional area (A c ). The luid and regenerator temperature variations over a steady-state cycle are the eventual output o the model (T (x,t) and T r (x,t)). These variations, coupled with the prescribed mass low rate and material properties, allow the calculation o various cycle perormance metrics such as the rerigeration load and the magnetic power requirement. These temperature variations are obtained by solving a set o coupled, partial dierential equations in time and space. The governing dierential equations are obtained rom dierential energy balances on the luid and the matrix. Figure illustrates a dierential segment o the luid with the various energy lows indicated. 3

5 heat transer: (, ) ( ) Nu Re Pr k T aa s c( T T r) dx enthalpy inlow: () ( T ) mth enthalpy outlow: mth () ( T ) + mth () ( T ) dx x axial dispersion LHS: k disp T Ac x axial dispersion RHS: T T kdisp Ac kdisp Ac dx x x viscous dissipation: () p mt x ρ dx energy storage: ρεau t c dx Figure. Dierential segment o entrained luid with energy terms indicated Ater some simpliication, the energy balance on the luid suggested by Fig. is: T h Nuk p m u disp c s c ( r ) + = ρ c ε x x x ρ t k A m a A T T A (1) The irst term in Eq. (1) represents conduction due to axial dispersion; the second term represents the change in the enthalpy carried by the luid; the third term is the convective heat transer between the luid and the regenerator material; the ourth term represents viscous dissipation in the luid, and the right side o the equation represents energy stored due to the heat capacity o the luid entrained in the matrix. Note that axial conduction through the luid is considered together with the axial conduction in the bed. Conduction in the luid may be non-negligible due to its relatively high thermal conductivity. However, the axial conduction is applied to the matrix and modeled using the concept o an eective static bed conductivity. Note that the dispersive conductivity is much higher than the conductivity o the luid whenever the luid is lowing. 4

6 Ater expanding the derivatives in Eq. (1) under the assumption that material properties are independent o pressure and substituting the deinition o the riction actor in terms o the pressure gradient, the energy balance becomes: T h T Nuk m u T k A m a A T T A () 3 disp c s c ( r ) + = ρ c ε x T x ρ Ac T t Assuming an incompressible luid, Eq. () can be simpliied to: T T Nuk m T k A mc a A T T A c (3) 3 disp c s c ( r ) + = ρ c ε x x ρ Ac t Figure 3 illustrates a dierential segment o the regenerator material with the various energy lows indicated: heat transer: (, ) ( ) Nu Re Pr k T aa s c( T T r) dx axial conduction LHS: T r ke Ac x axial conduction RHS: Tr Tr ke Ac ke Ac dx x x magnetic work transer: M Ac( 1 ε ) µ oh dx t energy storage: ρac( 1 ε) ur dx t Figure 3. Dierential segment o regenerator with energy terms indicated 5

7 The energy balance suggested by Figure 3 is: Nu k M T u d t x t r r as As( T Tr) + Ac ( 1 ε ) µ oh + ke Ac = ρ r Ac ( 1 ε) h (4) The magnetic work term is grouped with the internal energy to obtain: ( v M) Nu k T r u r R as As( T Tr) + ke Ac = A c ( 1 ε) ρ r µ oh x t t (5) The right hand side o Equation (5) is the dierence between a dierential change in internal energy and a dierential work transer; this dierence must be equal to a dierential heat transer, which is related to a change in entropy. Thereore, assuming the magnetization and demagnetization are reversible, Equation (5) may be rewritten according to: Nu k T s d x t r r as As( T Tr) + ke Ac = A c ( 1 ε) ρr Tr h (6) The change in regenerator entropy is divided into temperature and magnetic ield driven components in order to yield the inal, regenerator governing equation: Nu k µ H d x H t t Tr sr o Tr as As( T Tr) + ke Ac = A c ( 1 ε) ρr Tr + Ac( 1 ε) ρr cµ oh h µ o T (7) The luid is assumed to enter the matrix at the temperature o the associated heat reservoir, providing the required spatial boundary conditions: ( ) ( 0 ) () ( ) i m t 0 then T x =,t = T i m t <0 then T x = L,t = T H C (8) The governing equations are integrated orward in time using a spatially implicit technique. A periodic steady state is achieved when the total energy change o the bed material ( U r ) and luid entrained in the bed ( U ) between the end o cycle k and the end o the previous cycle, k-1, normalized by the dierence 6

8 between the maximum and minimum energy stored in the regenerator over the cycle is within a convergence tolerance (tol). U U max + U + U r min < tol steady state (9) where the change in luid energy is calculated by integrating the absolute value o energy change over the length o the bed. L U = ρεa u u dx (10) c k k 1 0 and the change in regenerator energy is ( 1 ) r r 1 r r c k k 0 L U = ρ ε A u u dx (11) and the energy stored in the luid is L U = ρεa u dx (1) c 0 where T re is an arbitrarily chosen reerence temperature. The energy o the regenerator material is calculated in same manner as the luid energy. U max is deined as the maximum sum o the luid and regenerator energy at a given time step in the cycle and U min is the minimum sum o luid and regenerator energy. The numerical solution or the luid and regenerator temperature is obtained over a spatial grid that extends rom 0 to L as shown in Figure 4. 7

9 L x, 1, i, m r, 1 r, i r, m Figure 4. Numerical grid used or luid and regenerator temperature solutions The axial location o each luid and regenerator temperature node (x i ) is given by: 1 L x i = i i=1..m m (13) where i is the axial subscript and m is the total number o axial control volumes that are used. The cycle time is discretized by: τ t j = j j=0..n (14) n where j is the temporal subscript and n is the total number o time steps that are used. Initial values or the temperatures at each spatial node ( T ri, 1 and T i, 1 ) can be assigned arbitrarily. One possibility is an assumed linear temperature proile, although other options are explored to speed convergence. xi Tri, 1 = TH ( TH T C) i=1..m (15) L xi Ti, 1 = TH ( TH T C) i=1..m (16) L The changes in luid and regenerator properties over a small time step are neglected so that the temperatures at time step i+1 are obtained using the discretized governing equations with constant 8

10 properties that are evaluated at time step i. The luid energy balance is discretized and written or each control volume. Nu k a A L T + T T + T ( T 1 T 1) m + + ( t ) c i,j i,j s c i,j+1 i-1,j+1 i+1,j+1 i,j+1 ri,j i,j + j i,j m ( j) c h 3 i,jm t L m m + + k A T T + k A T T ρ A d m L L ( ) ( ) disp i,j c i-1,j+1 i,j+1 disp i,j c i+1,j+1 i,j+1 (17) Ln = ρ Acεc T i,j+1 T i,j i =..m m τ ( ) 1 where Nu i, j is the Nusselt number based on the luid temperature values in the node, (, ) Nu = Nu Re Pr (18) i, j i, j i, j Re is the Reynolds number or the luid computed using the luid temperature and the ree low velocity and Pr i,j is the Prandtl number o the luid: Re i,j c ( j) m t = A µ i,j (19) Pr i,j c i,jµ i,j = (0) k i,j The riction actor ( i, j) in Equation (14) is evaluated in terms o the local Reynold s number which depends upon the luid temperatures within the node: ( ) = Re (1) i, j i, j The boundary conditions or the luid temperature governing equation are that luid that enters at either edge o the regenerator has the temperature o the corresponding reservoir and that the edges o the bed are adiabatic with respect to dispersive heat transer. At the boundaries o the regenerator bed, the energy 9

11 balance depends upon the luid low direction. The discretized luid equations at the hot end o the regenerator bed are: ( j ) i m t 0 then Nu k a A L T + T ( T + T + ) + m ( t ) c T 1,j 1,j s c,j+1 1,j+1 r 1,j 1 1,j 1 j 1,j H m ( j) 3 1,jm t L m Ln disp 1,j c,j+1,j+1 c,j+1,j ρ Ac m L mτ ( 1 ) ρ ε ( 1 1 ) + + k A T T = A c T T or i= 1 ( j ) i m t < 0 then Nu k a A L T + T 3 1 ( T + T + ) m ( t ) c T T 1,j 1,j s c,j+1 1,j+1 r 1,j 1 1,j 1 j 1,j 1,j+1,j+1 m ( j) c h 3 1,jm t L m Ln + + k A T T = A c T T ρ A d m L mτ ( ) ρ ε ( ) disp 1,j c,j+1 1,j+1 c 1,j+1 1,j or i =1 (a) (b) Note that, as shown in Figure 4, the temperature o each node is evaluated at the center o the node. Thereore, the temperature o the luid exiting the hot end o the bed is approximated by extrapolating the temperatures in nodes 1 and. For node m (the cold end o the bed), the energy balances are: ( j ) i m t 0 then Nu k a A L T + T 3 1 ( T + T + ) + m ( t ) c T T m,j m,j s c m,j+1 m- 1,j+1 rm,j 1 m,j 1 j m,j m,j+1 m- 1,j+1 m ( j) c h 3 m,jm t L m Ln + + k A T T = A c T T ρ A d m L mτ ( j ) i m t < 0 then ( j) ( 1 ) ρ ε ( ) disp m,j c m-,j+1 m,j+1 c m,j+1 m,j Nu k a A L T + T ( T + 1 T + 1) m ( t ) c T m,j m,j s c m,j+1 m- 1,j+1 rm,j m,j j m,j C m c h 3 or i=m m,jm t L m Ln + + kdisp m,j Ac ( T m- 1,j+1 T m,j+1 ) = ρ Acεc ( T m,j+1 T m,j ) or i=m ρ A d m L mτ (3a) (3b) 10

12 Collecting like terms in equations (17) leads to: L n as AcL m c i,j m T i,j+ 1 ερ Ac c i,j + Nu i,j k i,j + kdispac + T i 1,j+ 1 m ( tj ) kdispac m τ m L L ci,j m Nui,jk i,j L n + Ti + 1,j+ 1 m ( tj) kdispac + Tri,j+ 1 as Ac = Ti,j ερac c i,j L d h m τ ( ) 3 j * i,j m t L + i =..m 1 ρ A d m c h (4) Collecting terms or the hot end energy balance in equations (a) and (b) yields: ( j ) i m t 0 then L n ci,j as AcL m ci,j m Ti,j+ 1 ερac c i,j+ m ( tj) + Nui,jki,j + kdispac + Ti + 1,j+ 1 m ( tj) kdispac m τ m L L 3 ( j ) * Nu i,jk i,j L n i,j m t L + Tri,j+ 1 as Ac = T i,j ερ Ac c i,j + d h m τ + m ( t ) c T ρ Ac m i = 1 (5a) ( j ) i m t < 0 then * Nu i,jki,j L n i,j ri,j+ 1 s c i,j c i,j d ερ h m τ ρ ( j ) 3 j i,j H L n as AcL m m Ti,j+ 1 ερac c i,j m ( tj) ci,j+ Nui,jki,j + kdispac + Ti + 1,j+ 1 m( tj) ci,j kdispac m τ m L L + T a A = T A c + i = 1 (5b) m t L A d m c h Collecting terms or the cold end luid energy balance in equations (3a) and (3b) yields 11

13 ( j ) i m t 0 then Ln as AcL m Ti,j+ 1 ερac c i,j+ m ( tj) ci,j+ Nui,jki,j + kdispac mτ m L m Nu i,jk i,j + T i 1,j+ 1 m ( tj) c i,j kdispac + Tri,j+ 1 as Ac L Ln = Ti,j ερac c i,j + mτ ( j ) i m t < 0 then * i,j ( j ) c 3 m t L ρ A d m h i = m Ln c i,j as AcL m Ti,j+ 1 ερac c i,j m ( tj) + Nui,jki,j + kdispac mτ m L ci,j m Nui,jk i,j + T i 1,j+ 1 m ( tj) kdispac + Tri,j+ 1 as Ac L ( ) 3 j * Ln i,j m t L = Ti,j ερac c i,j m( tj) ci,jtc + mτ ρ Ac m i = m (6a) (6b) The regenerator energy balances are likewise discretized and written or each control volume: Nu i,jki,j L L a A ( T + 1 T + 1) + A ( 1 ε) ρ c d m m h ( 1 ε) s s ri,j i,j c r µ oh i,j mke i, j Ac mke i, j Ac + Tri,j 1 T ri 1,j 1 Tri,j 1 T ri + 1,j+ 1 L + = L ρ s µ H t + t r o j+ 1 j Ac rtri,j x i, µ oh t i,j ( ri,j+ 1 ri,j) T T n τ i =..m 1 (7) Note that the 3 rd and 4 th terms in Eq. (7) represent conduction to the neighboring control volumes on the let- and right-hand sides, respectively. Collecting like terms leads to: 1

14 Nui,j k i,j Ln m ke i,j A c Tri,j+ 1 as As + Ac ( 1 ε) ρr cµ ohi,j + mτ L mke i,j A c mke i,j A c Nui,j k i,j + Tri 1,j+ 1 + Tri + 1,j+ 1 + T i,j+ 1 as As L L dp t = A T x, i = 1..m sr µ oh j+ 1 j c ( 1 ε) ρr ri,j i + Ac ( 1 ε) ρr cµ oh i,jtri,j µ oh t i,j + t Ln mτ (8) The hot and cold ends o the regenerator are assumed adiabatic. Neglecting conduction at the edge o the bed, the energy balance at the hot end is: Nu 1,jk 1,j L L a A ( T + T + ) + A ( 1 ε) ρ c d m m h s s r 1,j 1 1,j 1 c r µ oh 1,j ( r 1,j+ 1 r 1,j) T T n mk A s µ H t + t + = ( 1 ε) e 1, j c r o j+ 1 j Tr 1,j+ 1 Tr,j+ 1 Ac ρrtr 1,j x 1i, L µ o H t 1,j τ (9) Neglecting conduction at the cold end, the energy balance at the cold end is: Nu m,jkm,j L L a A ( T + 1 T + 1) + A ( 1 ε) ρ c d m m h s s r m,j m,j c r µ oh m,j ( r m,j+ 1 rm,j) T T n mk A s µ H t + t + = ( 1 ε) e m, j c r o j+ 1 j Trm,j+ 1 Trm 1,j+ 1 Ac ρrtrm,j x 1i, L µ oh t m,j τ (30). Numerical Solution Algorithm Equations (4)-(6) and (8)-(30) orm a system o linear equations in terms o each o the nodal regenerator and luid temperatures that are shown in Figure 4 at one step orward in time. These equations are solved using a sparse matrix decomposition algorithm in order take a spatially implicit but temporally explicit step orward in time. This time step solution process is repeated in order to determine the luid and regenerator temperatures at all spatial nodes and or each time step over an entire cycle. At 13

15 the end o each cycle, the change in energy in the regenerator and luid is evaluated by perorming the integration in equations (10) and (11) numerically and compared to the total energy in the luid at the end o the cycle. When the absolute change in energy o the regenerator rom cycle to cycle is within a speciied tolerance, shown in Eq. (9), steady state has been achieved. This model is implemented in MATLAB. The assumptions used to derive the numerical model were described as the model was derived in the previous section and are summarized below: the heat transer luid is incompressible; thereore the mass low rate does not vary spatially within the matrix and the mass o luid entrained in the matrix is constant, the bed geometry is uniorm; no spatial gradients exist in the bed characteristics such as particle diameter, porosity, etc., the luid low is one-dimensional; low maldistribution eects are neglected, and the luid low is balanced, the magnetization and demagnetization processes are modeled as being internally reversible with no hysteresis or temperature gradients (note that this assumption is subsequently revisited and ultimately considered via a correction actor). 3. Veriication o Model There are no general analytical solutions to the regenerator equations presented above. However, in the limit o constant properties, no entrained luid heat capacity and no axial conduction, a published solution or the thermal eectiveness (ε) o a conventional, passive regenerator (i.e., one with no magnetocaloric eect) subjected to a stepwise mass low rate variation (with the shape shown in Figure 5) is available. 14

16 m m τ t Figure 5. Mass low rate variation or an idealized regenerator According to Rohsenow et al. (1998), the thermal eectiveness (ε t ) or a regenerator with constant material properties is deined as: t τ / ( ) () ( =, ) mc ( T T ) c m 0 t TH T x L t dt ε H C (31) where m is the magnitude o the mass low rate unction. The typical variables used to characterize this problem are the number o transer units (NTU, sometimes also reerred to as the reduced length o the regenerator) and the utilization ratio (U, the inverse o the matrix capacity rate ratio, Ackermann (1997)). NTU Nu k s c p a L A d mc (3) U mc τ = A L( 1 ε) ρ c µ c r oh (33) Dragutinovic and Baclic (1998) present tables or the ε t as a unction o NTU and U in this limit. The numerical model can be veriied against these solutions by: 1. setting all luid properties (c, k, ρ, and µ ) equal to constants,. setting the partial derivative o entropy with respect to magnetic ield equal to zero, 15

17 3. setting the remaining regenerator properties ( c µ and ρ r ) equal to constants, 4. setting the eective thermal conductivity o the matrix (k e ) and dispersion (k disp ) equal to zero, 5. setting the riction actor () equal to zero, o H 6. setting the speciic surace area o the regenerator (a s ), particle diameter (d p ) and bed size (A c and L) equal to constants, 7. applying the unctional orm o the mass low rate shown in Figure.6 or a ixed cycle duration ( τ ), τ mt () = sign t m (34) 8. and setting the porosity (ε) to zero in order to speciy zero entrained luid heat capacity. By varying the Nusselt number (Nu ) and mass low rate ( m ), it is possible to vary NTU and U. The numerical model was implemented under these conditions using a grid with 100 spatial control volumes (m = 100) and 3000 time steps (n = 3000). The results are illustrated in Figure 6. Notice the excellent agreement between the published and predicted results, veriying the accuracy o the numerical model in this limit. The results in Figure 6 are plotted in the region were eectiveness is greater than 0.9 in Figure 7. 16

18 Figure 6. Numerical model predictions and published results or ε t as a unction o NTU and various values o U in the ideal regenerator limit Figure 7. Numerical model predictions and published results rom Figure 6 in the region ε t >

19 Nomenclature A c cross-sectional area (m ) a s speciic surace area (m /m 3 ) c speciic heat capacity (J/kg-K) d p particle diameter (m) riction actor k thermal conductivity (W/m-K) k e eective static thermal conductivity o regenerator and luid (W/m-K) k disp thermal conductivity o the luid due to axial dispersion (W/m-K) i spatial subscript j temporal subscript L length (m) m number o axial control volumes used in numerical solution m mass low rate (kg/s) M magnetic intensity (A/m) n number o steps used in numerical solution NTU number o transer units Nu Nusselt number p pressure (Pa) Pr Prandtl number q average heat transer rate (W) Re Reynolds number s entropy (J/kg-K) t time (s) tol relaxation tolerance (K) T temperature (K) T Curie Curie temperature (K) u internal energy (J/kg) U utilization actor v speciic volume (m 3 /kg) x axial position (m) Greek ε porosity o matrix ε t thermal eectiveness µ viscosity (N-s/m ) µ o H applied ield (Tesla) ρ density (kg/m 3 ) τ cycle duration (s) Subscripts C H cold or rerigeration temperature luid hot or heat rejection temperature 18

20 r regenerator material Reerences R. A. Ackermann, 1997, Cryogenic Regenerative Heat Exchangers, Plenum Press, New York. G. D. Dragutinovic, and B. S. Baclic, 1998, Operation o Counterlow Regenerators, Computational Mechanics Inc., Billerica, MA. W. M. Rohsenow, J. P. Hartnett and Y. I. Cho, 1998, Handbook o Heat Transer, McGraw-Hill, New York, NY. 19