Heat Load Analysis on LHe e-bubble Chamber Cryostat

Heat Load Analysis on LHe e-bubble Chamber Cryostat Yonglin Ju Nevis Laboratories, Columbia University, NY 10533 1. Heat load rate per surface area @4K 2. LHe boil-off rate (heat load) @4K 3. Heat load rate per surface area @80K LN2 boil-off rate (heat load) @ 80K 5. Heat load to the e-bubble chamber 6. Vapor cooling rate 1. Heat load rate per surface area @ 4K (1) Since the vacuum chamber is evacuated by the turbomolecular pumping system to less than 10-5 torr, even to 10-7 torr by cryopumped by liquid helium, making convective heating negligible. Radiative and conductive heat leaks can be evaluated separately as following (2) For an ordinary metal surface (Al or copper), with average emissivity of 0.05, radiating at 300K to a similar surface at 4K, the radiant heat transferred is a sizeable qv = 11.72 W/m 2, which will boil away 1.65 cm 3 of LHe per hour. However, by interposing a radiation shield cooled to LN2 temperature between the two surfaces, the heat leak to the 4K surface will be reduced by the factor of (300/80) 4 =200, qv = 0.06 W/m 2 (3) Conductive heat losses come from the HVF cable, e-bubble central tube, LHe vessel neck, LHe needle valve tube, and LHe level sensor, which are calculated based on their cross-sectional area, the length and the thermal conductivity averaged over the temperature difference across the conductor T_room 300 T_shield 80 T_n2 77.3 T_he 22 [K] σ 5.67. 10 8 [W/ K 4 m 2 ] [Stefan-Boltzmann constant] 0.05 [Average emissivity of Aluminum and copper] ε_ss 0.1 [Average emissivity of Stainless Steel] Keff_300_80 12.2 [W/mK ] Keff_300_4 10.3 [W/mK ] Keff_80_4 6 [W/mK ] Q_cond Keff_80_ T_shield T_he A. L Q_300_80 σ. T_room4 T_shield 4 1 1 Q_300_80 = 11.717 [W/m 2 ] Q_300_4 σ. T_room4 T_he 4 1 1 Q_300_80 = 0.558 [W/m 2 ] [with MLI, 20 layers] ( 20 1) Q_300_4 = 11.776 [W/m 2 ] Q_80_4 σ. T_shield 4 T_he 4 1 1 factor1 Q_80_4 = 0.06 [W/m 2 ] Q_300_4 Q_80_4 factor1 = 197.755 1

A. LHe 4K e-bubble chamber Pv 101.3. [Pa] Tv 22 [K] ρfv 12955 [kg/m 3 ] ρgv 16.8407 [kg/m 3 ] hfv 5.740710. 3 [J/kg] hgv 198110. 3 [J/kg] hfgv hgv hfv hfgv = 2.072. 10 4 [J/kg] Dv 0.108 [m] Hv 0.170 [m] Hfv 0.105 [m] [Hight of LHe in e-bubble chamber] Av1 Av 4 Dv2 Av1 = 9.161. [m 2 ] Av2 Dv. Hv Av2 = 0.058 [m 2 ] 2. Av1 Av2 Av = 0.076 [m 2 ] [Cold surface area of e-bubble chamber] Vfv. 4 Dv2. Hfv Vfv = 9.619. 10 4 [m 3 ] Ve_he Vfv. 1000 Ve_he = 0.962 [L] [Volume of LHe in e-bubble chamber] Assuming heat load to the 4K e-bubble chamber: qv 0.06 [W/m 2 ] Qev Av. qv Qev = 56. [W] ############################################################################################## B. LHe reservoir Pr 101.3. [Pa] Tr 22 [K] hfgr 2.072. 10 4 [J/kg] ρfr 12955 [kg/m 3 ] Dr 12. 25. Dr = 0.305 [m] Hr 21.5. 25. Hr = 0.546 [m] Dz 0.020 [m] Dn 5. 25. Dn = 0.127 [m] Hn 20. 25. Hn = 0.508 [m] A1 A2 4 Dr2 Dn 2 A1 = 0.06 [m 2 ] A3 Dn. Hn A3 = 0.203 [m 2 ] 4 Dr2 Dz 2 A2 = 0.073 [m 2 ] A4 Dr. Hr A4 = 0.523 [m 2 ] Ar A1 A3 A4 Ar = 0.786 [m 2 ] [Cold surface of the LHe reservoir] Vr 4 Dr2 Dz 2. Hr Vr = 0.04 [m 3 ] Vr. 1000 = 39.675 [L] [Volume of LHe in LHe reservoir] Vn. 4 Dn2 Dz 2. Hn Vn = 6.276. [m 3 ] Vn. 1000 = 6.276 [L] [Volume of LHe in neck space] Vr_he ( Vr Vn). 1000 Vr_he = 45.951 [Total volume of LHe in LHe reservoir] Assuming heat load to the LHe reservoir: qr 0.06 [W/m 2 ] Qrev Ar. qr Qrev = 0.047 [W] ############################################################################################### C. 4K shield Ds 9. 25. Ds = 0.229 [m] Hs ( 26 1.25 1.0). 25. Hs = 0.603 [m] As1 4 Ds2 As1 = 0.041 [m 2 ] As2 Ds. Hs As2 = 0.433 [m 2 ] As As1 As2 As = 0.474 [m 2 ] [Cold surface of the 4K shield] Assuming heat load to the 4K shield: qs 0.06 [W/m 2 ] Qshield As. qs Qshield = 0.028 [W] 2

D. HVF cable heat conduction d_tef 3. [m] d_ss 1.0. [m] 0.2 [m] N 10 A_ss. 4 d_ss 2 A_ss = 7.85 10 7 [m^2] A_tef. 4 d_tef 2 d_ss 2 A_tef = 6.283. 10 6 [m^2] Teflon insulator Ktef_80 0.22 [W/mK] Ktef_4 0.045 [W/mK] Ktef_80_4 10 [W/m] [thermal conductivity intergal] SF_tef A_tef SF_tef = 3.142. [SF: shape factor] Qc_tef SF_tef. Ktef_80_4 Qc_tef = 398. 10 4 [W] Stainless steel Kss_80 7.8 [W/mK] Kss_4 0.3 [W/mK] Kss_80_4 349 [W/m] [thermal conductivity intergal] SF_ss Qc_ss A_ss SF_ss = 3.927. 10 6 [m] SF_ss. Kss_80_4 Qc_ss = 1.371. [W] Qcable N. ( Qc_tef Qc_ss ) Qcable = 0.018 [W] ################################################################################################ E. Heat conduction through LHe v essel neck d_neck 5.5. 25. d_neck = 0.14 [m] L_neck 25. 25. L_neck = 0.318 [m] 2 V_neck. 4 d_neck 2 Dz 2. L_neck V_neck = 767. [m 3 ] δ_neck 0.2. [m] S_neck d_neck. δ_neck S_neck = 8.778. 10 5 [m 2 ] SF δ_neck. d_neck L_neck SF = 2.765. 10 4 [m] [SF: shape factor] Kss_80_4 349 [W/m] [thermal conductivity intergal] Qnc SF. Kss_80_4 Qnc = 0.096 [W] ################################################################################################ F. Radiation in LHe vessel neck d_neck = 0.14 [m] L_neck = 0.318 [m] A_neck d_neck. L_neck A_neck = 0.139 [m 2 ] T_shield = 80 T_he 22 Qnr1 Qnr T_n2 4 T_he 4 σ. Qnr1 = 0.107 [W/m 2 ] 1 ( 1 ε_ss) ε_ss ε_ss Qnr1. A_neck Qnr = 0.015 [W] 3

G. Heat conduction through e-chamber central transfer tube d_tt 0.5. 25. δ_tt 0.02. 25. L_tt ( 20.0 5). 25. d_tt = 0.013 [m] δ_tt. 1000 = 0.508 [mm] L_tt = 0.635 [m] Kss_300 19 [W/mK] Kss_4 0.3 [W/mK] Kss_300_4 3090 [W/m] [thermal conductivity intergal] SF_tt d_tt 2 ( d_tt 2. δ_tt ) 2 L_tt SF_tt = 3.06 Qtt SF_tt. Kss_300_4 Qtt = 0.095 [W] Minimum wall thickness P δ 150. 17 1.013. 10 5 [Pa] σr 1.32. 10 9 [Pa] P.( d_tt 2. δ_tt ) 2. σr 0.8. P δ = 573. 10 6 [m] δ δ. 1000 δ = 573. [mm] ############################################################################################## H. Heat conduction through needle valve stem d_nv_shell 8. [m] δ_nv_shell 0.2. [m] d_nv_stem 6. [m] L_nv 16.0. 25. L_nv = 0.406 [m] Needle valve stem Ktef_300 0.30 [W/mK] Ktef_4 0.04 [W/mK] Ktef_300_4 70.2 [W/m] [thermal conductivity intergal] SF_nv_stem ( d_nv_stem )2 L_nv SF_nv_stem = 6.957. Qnv_stem SF_nv_stem. Ktef_300_4 Qnv_stem = 88 [W] Needle valve shell Kss_300_4 3090 [W/m] [thermal conductivity intergal] SF_nv_shell d_nv_shell. δ_nv_shell L_nv SF_nv_shell = 1.237. Qnv_shell SF_nv_shell. Kss_300_4 Qnv_shell = 0.038 [W] Qnv Qnv_stem Qnv_shell Qnv = 0.043 [W] ############################################################################################## I. Heat conduction through level sensor d_sen_od 8. [m] d_sen_id 6.0. [m] L_sen 16.0. 25. L_sen = 0.406 [m] LHe level sensor Ktef_300_4 70.2 [W/m] [thermal conductivity intergal] SF_sen d_sen_od 2 d_sen_id 2 L_sen SF_sen = 5.411. Qsen SF_sen. Ktef_300_4 Qsen = 3.799. [W] 4

Total heat load to the summary to the LHe reservoir and the 4K heat shield Qev = 56. Qrev = 0.047 Qshield = 0.028 Qcable = 0.018 Qnc = 0.096 Qnr = 0.015 Qtt = 0.095 Qnv = 0.043 Qsen = 3.799. Qtotal Qev Qrev Qshield Qcable Qnr Qnc Qnv Qsen Qtt Qtotal = 0.351 [W] 2. LHe boil-off rate @ 4K m_4k Qtotal hfgr m_4k = 1.695. 10 5 [kg/s] m_4k. 1000 = 0.017 [g/s] m_4k_v m_4k ρfr Vr_he = 45.951 [L] T_hour_He T_day_He T_hour_He 24 3. Heat load per surface area @80K m_4k_v= 1.356. 10 7 [m 3 /s] m_4k_v. 1000. 3600 = 0.488 [L/h] Vr_he m_4k_v. 1000. 3600 [hr] T_hour_He = 91 [hr] [day] T_day_He = 3.921 [day] Pn 101.3. [Pa] Tn2 77.3 [K] hfgn 198.9. [J/kg] ρfn 806.804 [kg/m 3 ] Dno 1 25. Dno = 0.356 [m] Dni 6. 25. Dni = 0.152 [m] Hnz 20. 25. Hnz = 0.508 [m] Vn 4 Dno2 Dni 2. Hnz Vn = 0.041 [m 3 ] Vn. 1000 = 41.185 [L] [Volume of liquid in the LN2 reservoir] Hn1 ( 21.5 1 1.25). 25. Hn1 = 0.603 [m] [The upper section of LN2 shield] Hn2 ( 26 1.25 1.0). 25. Hn2 = 0.603 [m] [The lower section of LN2 shield] Dn1 13. 25. Dn1 = 0.33 [m] Dn2 11. 25. Dn2 = 0.279 [m] Hfn 10. 25. Hfn = 0.254 [m] An1 2. Dno 2 Dni 2 Dno. Hnz Dni. Hfn An1 = 0.851 [m 2 ] [Cold surface of LN2 reservoir] 4 An2 An3 An4 Dn1. Hn1 An2 = 0.626 [m 2 ] [Cold surface of upper section of 80K shield] Dn2. Hn2 An3 = 0.53 [m 2 ] 4 Dn22 An4 = 0.061 [m 2 ] An3 An4 = 0.591 [m 2 ] [Cold surface of lower section of 80K shield] An An1 An2 An3 An4 An = 2.068 [m 2 ] [Total cold surface eara @ 80K] A. Heat radiation on the LN2 reservoir and the 80K heat shield (1) Heat radiation from 300K surface to the LN2 reservoir and the 80K heat shield (with MLI, 20 layers): T_room = 300 T_shield 80 0.05 ε_ss 0.1 5

Q_nrs1 σ. T_room4 T_shield 4 Q_nrs1 = 15.757 [W/m 2 ] 1 ( 1 ε_ss) ε_ss Q_nrs1 = 0.75 [W/m 2 ] [with MLI, 20 layers] ( 20 1) Q_nrs Q_nrs1.( An1 An2 An3) Q_nrs = 1.506 [W] 21 (2) Heat radiation from 300K botton surface to the 80K heat shield bottom (without MLI): Dno = 0.356 Dn2 = 0.279 Ano 4 Dno2 Ano = 0.099 [m 2 ] An4 = 0.061 [m 2 ] Q_nrb2 σ. T_room4 T_shield 4 Q_nrb2 = 17.88 [W/m 2 ] 1 ( 1 ε_ss). An4 ε_ss Ano Q_nrb Q_nrb2. An4 Q_nrb = 1.096 [W] Q_nr Q_nrs Q_nrb Q_nr = 2.602 B. Heat radiation on the 80K heat shield through the glass windows ( 5 pieces ) d_w1 1.97. 25. d_w1 = 0.05 [m] ε_glass 0.90 d_w2 2.50. 25. d_w2 = 0.064 [m] A_w1 Q_wr1 Q_wr 4 d_w12 A_w1 = 1.966. A_w2 4 d_w22 A_w2 = 3.167. [m 2 ] T_room 4 T_shield 4 σ. [W/m 2 ] 1 ( 1 ε_glass ) A_w1 Q_wr1 = 387.209 ε_glass ε_glass A_w2 Q_wr1. A_w1 Q_wr = 0.761 [W] Q_wr. 5 = 3.807 [W] C. Heat conduction through the pumping line d_pump 0.75. 25. d_pump = 0.019 [m] δ_pump 0.0 25. δ_pump = 1.016. [m] L_pump 25. 25. L_pump = 0.318 [m] 2 Keff_300_80= 12.2 Kss_300_80 Keff_300_80. ( T_room T_shield ) Kss_300_80 = 2.68 SF_pump d_pump 2 ( d_pump 2. δ_pump ) 2 L_pump SF_pump = 1.813. 10 4 [m] Q_pump SF_pump. Kss_300_80 Q_pump = 0.487 [W] D. Heat conduction through the LN2 fill and vent tube d_fill 0.75. 25. d_fill = 0.019 [m] δ_fill 0.0 25. δ_fill = 1.016. [m] L_fill 5 25. L_fill= 0.127 [m] Keff_300_80= 12.2 Kss_300_80 Keff_300_80. ( T_room T_shield ) Kss_300_80 = 2.68 SF_fill d_fill 2 ( d_fill 2. δ_fill) L_fill SF_fill = 532. 10 4 [m] Q_fill SF_fill. Kss_300_80 Q_fill = 1.217 [W] 6

E. Heat conduction through the HVF cable d_tef 3. [m] d_ss 1.0. [m] 0.20 [m] N 10 A_ss 4 d_ss 2 A_ss = 7.85 10 7 [m^2] A_tef 4 d_tef 2 d_ss 2 A_tef = 6.283. 10 6 [m^2] Teflon insulator Ktef_300_80 56.2 [W/m] [thermal conductivity intergal] SF_tef A_tef SF_tef = 3.142. [SF: shape factor] Qc_tef SF_tef. Ktef_300_80 Qc_tef = 1.766. [W] Stainless steel Kss_300_80 = 2.68 [W/m] [thermal conductivity intergal] SF_ss A_ss SF_ss = 3.927. 10 6 [m] Qc_ss SF_ss. Kss_300_80 Qc_ss = 0.011 [W] Q_cable N. ( Qc_tef Qc_ss ) Q_cable = 0.123 [W] F. Heat conduction through LHe v essel neck d_neck 5.5. 25. d_neck = 0.14 [m] L_neck 25. L_neck = 0.102 [m] V_neck. 4 d_neck 2 Dz 2. L_neck V_neck = 1.525. [m 3 ] δ_neck 0.2. [m] S_neck d_neck. δ_neck S_neck = 8.778. 10 5 [m 2 ] SF δ_neck. d_neck L_neck SF = 8.639. 10 4 [m] [SF: shape factor] Kss_300_80 = 2.68 [W/m] [thermal conductivity intergal] Q_nc SF. Kss_300_80 Q_nc = 2.319 [W] Total heat load to the LN2 reservoir and the 80K heat shield Q_nr = 2.602 Q_wr = 0.761 Q_pump = 0.487 Q_fill = 1.217 Q_cable = 0.123 Q_nc = 2.319 Q_total Q_nr 5. Q_wr 2. Q_pump 3. Q_fill Q_cable Q_nc Q_total = 13.474 [W] LN2 boil-off rate @80K The evaporation rate of N2 in the reservoir: m_80 m_80_v Q_total hfgn m_80 ρfn Vn. 1000 = 41.185 [L] T_hour_N2 m_80 = 6.77 10 5 [kg/s] m_80. 1000 = 0.068 [g/s] m_80_v= 8.396. 10 8 [m 3 /s] m_80_v1000.. 3600 = 0.302 [L/h] Vn. 1000 m_80_v1000.. 3600 [hr] T_hour_N2 = 136.256 [hr] T_day_N2 T_hour_N2 24 T_day_N2 = 5.677 [day] 7

5. Heat load to the e-bubble chamber A Heat conduction through the support from He reservoir to EBC d_1 0.75. 25. d_1 = 0.019 [m] δ_1 0.065. 25. δ_1 = 1.651. [m] L_1 ( 5.75 1.75). 25. L_1 = 0.191 [m] Kss_4_2 0.375 Kss_3_4 0.202 SF_1 d_1 2 ( d_1 2. δ_1) L_1 SF_1 = 737. 10 4 [m] Q_1 SF_1. ( Kss_4_2 Kss_3_4 ) Q_1 = 8.195. 10 5 [W] ############################################################################################ B Heat conduction from needle-valve-output to EBC d_2 0.25. 25. d_2 = 6.35. [m] δ_2 0.0005 [m] L_2 0.2 [m] Kss_4_2 0.375 Kss_3_4 0.202 SF_2 d_2 2 ( d_2 2. δ_2) L_2 SF_2 = 595. Q_2 SF_2. ( Kss_4_2 Kss_3_4 ) Q_2 = 7.949. 10 6 [W] ############################################################################################# C Heat conduction from the pumping line to EBC d_3 0.5. 25. d_3 = 0.013 [m] δ_3 0.0005 [m] L_3 0.3 [m] Kss_4_2 0.375 Kss_3_4 0.202 d_3 2 ( d_3 2. δ_3) 2 SF_3 SF_3 = 6.388. L_3 Q_3 SF_3. ( Kss_4_2 Kss_3_4 ) Q_3 = 1.105. 10 5 [W] ############################################################################################ D Heat conduction through the HVF cables to EBC d_tef 3. [m] d_ss 1.0. [m] 0.3 [m] N 10 Ktef_4_2 0.043 Ktef_3_4 0.0123 A_tef 4 d_tef2 d_ss 2 A_tef = 6.283. 10 6 [m 2 ] Kss_4_2 0.375 Kss_3_4 0.202 A_ss = 7.85 10 7 [m 2 ] A_ss. 4 d_ss 2 Teflon insulator Stainless steel SF_tef A_tef SF_tef = 2.09 Q_4_tef SF_tef. ( Ktef_4_2 Ktef_3_4) Q_4_tef = 6.43. 10 7 [W] SF_ss A_ss SF_ss = 2.618. 10 6 [m] Q_4_ss SF_ss.( Kss_4_2 Kss_3_4 ) Q_4_ss = 529. 10 7 [W] Q_4 N. ( Q_4_tef Q_4_ss ) Q_4 = 1.096. 10 5 [W] 8

E Heat radiation to EBC Dv 0.108 [m] Hv 0.170 [m] Av1 Av 4 Dv2 Av1 = 9.161. [m 2 ] Av2 Dv. Hv Av2 = 0.058 [m 2 ] 2. Av1 Av2 Av = 0.076 [m 2 ] ε1 0.12 [Emissivity of stainless steel of EBC at 3.4K] ε2 0.062 [Emissivity of aluminum of 4K shield at 2K] T2 2 [K] T1 3.4 [K] As1 = 0.041 As2 = 0.433 Ashield 2. As1 As2 Ashield = 0.515 q_5 1 ε1 σ. T2 4 T1 4 1 ε2. Av ε2 Ashield q_5 = 9.528. 10 7 [W/m 2 ] Q_5 q_5. Av Q_5 = 7.242. 10 8 [W] ############################################################################################ F Heat load summary to EBC Q_1 = 8.195. 10 5 Q_2 = 7.949. 10 6 Q_3 = 1.105. 10 5 Q_4 = 1.096. 10 5 Q_5 = 7.242. 10 8 Q_EBC Q_1 Q_2 Q_3 Q_4 Q_5 Q_EBC = 1.12. 10 4 [W] Q_EBC. 1000 = 0.112 [mw] 6. Vapor cooling Hout 1215610. 3 [J/kg] Hin 5.8533910. 3 [J/kg] ρfv = 12955 [kg/m 3 ] Hout Hin = 2.007. 10 4 [J/kg] Q_EBC = 1.12. 10 4 [W] Q_guess 0.0. [W] m_vapor Q_EBC Hout Q_guess Hin m_vapor = 5.58. 10 9 [kg/s] m_vapor. 1000 = 5.58. 10 6 [g/s] m_vapor. 1000 = 466. 10 8 [L/s] ρfv m_vapor. 3600. 1000 = 1.608. 10 4 [L/hr] [Liquid consumption rate] ρfv ρg 0.1625 [kg/m 3 ] m_pump m_vapor. 760 m_pump = 7.908. 10 8 [m 3 /s] ρg 330 m_pump. 1000 = 7.908. 10 5 [L/s] m_pump. 1000. 3600 = 0.285 [L/hr] 9