Simulations of Fluid Dynamics and Heat Transfer in LH 2 Absorbers

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1 Simulations of Fluid Dynamics and Heat Transfer in LH 2 Absorbers Kevin W. Cassel, Aleksandr V. Obabko and Eyad A. Almasri Fluid Dynamics Research Center, Mechanical, Materials and Aerospace Engineering Department, Illinois Institute of Technology, Chicago, IL LH 2 Absorber Review Meeting Fermilab August 12-13, 2002

2 Introduction: Approaches to Heat Removal Two approaches under consideration: ➀ External cooling loop (traditional approach). Bring the LH 2 to the coolant (heat removed in an external heat exchanger). ➁ Combined absorber and heat exchanger. Bring the coolant, i.e. He,totheLH 2 (remove heat directly within absorber). He Cooling Tubes LH 2 Muon Beam Heater

3 Introduction (cont d) Advantages/disadvantages of an external cooling loop: + Has been used for several LH 2 targets (e.g. SLAC E158). + Easy to regulate bulk temperature of LH 2. + Is likely to work best for small aspect ratio (L/R) absorbers. May be difficult to maintain uniform vertical flow through the absorber. Advantages/disadvantages of a combined absorber/heat exchanger: + Takes advantage of natural convection transverse to the beam path. + Flow in absorber is self regulating, i.e. larger heat input more turbulence enhanced thermal mixing. + Is likely to work best for large aspect ratio (L/R) absorbers. More difficult to ensure against boiling at very high Rayleigh numbers.

4 Heat Exchanger Analysis Energy balance between LH 2 and coolant (He). Parameters: T i = coolant inlet temperature T o = coolant outlet temperature T LH2 = bulk temperature of LH 2 A = surface area of cooling tubes h LH2 = convective heat transfer coefficient of LH 2 h He = convective heat transfer coefficient of He x = thickness of cooling tube walls k w = thermal conductivity of cooling tube walls c p = specific heat capacity of He

5 Heat Exchanger Analysis (cont d) Rate of heat transfer: q = ( 1 h LH2 A(T o T i ) ) + x k w + 1 h He ln ( ) TLH2 T o T LH2 T i Mass flow rate of He: ṁ He = q c p (T o T i ). h He from appropriate correlation (flow through a tube). h LH2 and T LH2 from CFD simulations (no correlations for natural convection with heat generation).

6 Computational Fluid Dynamics (CFD) Features of the CFD Simulations: Provides average convective heat transfer coefficient and average LH 2 temperature for heat exchanger analysis. Track maximum LH 2 temperature (cf. boiling point). Determine details of fluid flow and heat transfer in absorber. Better understanding leads to better design!

7 CFD (cont d) Take 1: Results using FLUENT (M. Boghosian): Simulate one half of symmetric domain. Steady flow calculations. Heat generation via steady Gaussian distribution. Turbulence modeling (RANS) used for Ra Take 2: Results using COA code (A. Obabko and E. Almasri): Simulate full domain. Unsteady flow calculations. All scales computed for all Rayleigh numbers. Investigate startup behavior, e.g. startup overshoot in T max. Investigate possibility of asymmetric flow oscillations. Investigate influence of beam pulsing.

8 Formulation Properties and parameters: R = radius of absorber T w = wall temperature of absorber q (r) = rate of volumetric heat generation (Gaussian distribution) q = rate of heat generation per unit length ν = kinematic viscosity of LH 2 α = thermal diffusivity of LH 2 k = thermal conductivity of LH 2 β = coefficient of thermal expansion of LH 2

9 Energy equation: Governing Equations (T - ω - ψ formulation) T t + v T r r + v θ r Vorticity-transport equation: ω t + v ω r r + v θ r Streamfunction equation: ω θ 2 ψ r r T θ = 2 T r r v r = 1 r = Pr [ 2 ω r r +Ra R Pr ψ r ψ r 2 θ 2 ψ θ, T r + 1 r 2 2 T θ 2 + q(r) ω r + 1 ] 2 ω r 2 θ 2 [ sin θ T r + cos θ r = ω v θ = ψ r T θ ]

10 Formulation (cont d) Initial and boundary conditions: T = ω = ψ = v r = v θ =0 at t =0, T = ψ = v r = v θ =0 at r =1. Non-dimensional variables: r = r R, v r = v r R/α, v θ = v θ R/α, t = t R 2 /α, T = T T w q /k, ψ = ψ α, ω = ω α/r 2, q(r) = q (r) q /R 2 = 1 2πσ 2 e r 2 2σ 2, σ = σ R.

11 Formulation Non-Dimensional Parameters Prandtl Number: Pr = ν α Rayleigh Number: Ra R = GrP r = gr3 β q /k να ( = π ) 32 Ra MB Nusselt number: Nu R = h LH 2 R k ( = Nu ) MB 2

12 Results Flow Regimes Based on preliminary results, the following flow regimes are observed: Steady, symmetric solutions: Ra R Unsteady, asymmetric solutions: Ra R Steady, symmetric results for Ra R = (uniform heat generation): Streamfunction: Temperature: Vorticity: Y 0 Y X X

13 Steady, Symmetric Results (cont d) Nusselt number versus θ for Ra R = (uniform heat generation): Nu vs. θ:

14 Code Comparisons Average Nusselt Number ( Nu) Uniform heat generation (σ ) with Pr =1: Ra R Mitachi et al. 1 FLUENT 2 COA Code Mitachi et al. (1986, 1987) - Results shown are from numerical simulations which compared favorably with experiments. 2 From M. Boghosian s correlation for Pr =1.4, i.e. NuMB = Ra MB.

15 Steady, Symmetric Results: Ra R =1 10 8,σ =0.25 Streamfunction: Temperature: Vorticity: Y 0 Y X X

16 Unsteady, Asymmetric Results: Ra R =1 10 9,σ =0.25 t =0.2 Streamfunction: Vorticity: Y 0 Y X X Movies for streamfunction, temperature and vorticity (0 t 0.25).

17 Unsteady, Asymmetric Results: Ra R =1 10 9,σ =0.25 Asymmetric oscillation does not significantly influence wall heat transfer (e.g. Nu for left and right walls superimposed). t =2.0:

18 Unsteady Results High-Ra Startup Movie for Ra R = (σ =0.25). Nu vs. θ: t =

19 Current and Future Efforts Current and future work: Simulate Argonne test case and compare results. Determine critical Rayleigh number above which solutions are unsteady and asymmetric. Evaluate influence of σ, i.e. ratio of beam size to absorber size, on heat transfer. Obtain solutions at higher Rayleigh numbers (target Ra R ). Compare high-rayleigh number COA solutions (unsteady) with FLUENT results (steady RANS). Examine need for heater, e.g. to combat start-up overshoot. Investigate influence of pulsed beam on fluid dynamics and heat transfer. Note that at 15 Hz, one pulse corresponds to non-dimensional time units (cf. t =10 8 ).

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