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Cooling Air Flow StagesColor key:
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Contact: jhsilber@lbl.govINPUT
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CALCULATED
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Version historyLINKED
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v1 - June 24, 2025 - initial version
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v2 - June 27, 2025 - fixed corrugation area calc
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v3 - June 30, 2025 - fixed factor 1/2 in ΔP₂₃ₙ calc
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v4 - July 15, 2025 - 5 stages, cross-section totals
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v5 - Sept 8, 2025 - 6 stages, named, including exhaust stage
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v6 - Oct 21, 2025 - disk and barrel stave geometry details
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v7- Oct 21, 2025 - added module power loads
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Physical constants and unit conversions
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standard dynamic viscosity (air)μ₀Pa*s1.72E-05
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standard temperatureT₀K273
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specific gas constant (air)RJ/(kg*K)287
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heat capacity ratio (dry air @ 20C)γ-1.400
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specific heat capacity (dry air @ 20C)cₚ = R γ / (γ - 1)J/(kg*K)1004.5
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isentropic temperature exponentk = (γ-1)/γ-0.286
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critical pressure ratio (choked flow)βₓ = Pₓ / P = [2/(γ+1)]^[1/k]-0.528
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infinity∞-1E+99
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cfm per m³/scfm_per_m3s-2118.88
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Disk geometry
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corrugation pitchxcmm34.7
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corrugation heighthcmm6
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corrugation cross-sectional areaAc ~ xc * hc / 2mm²104.1
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corrugation hydraulic diameterDc = √(4 Ac / π)mm11.5
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large disk diameter (D1-D4)Dd14mm870
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small disk diameter (D0)Dd0mm488
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num large disks (per side)nd14-4
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num small disks (per side)nd0-1
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avg disk diameter (areally weighted)
Dd = √((Dd14²*nd14+Dd0²*nd0)/(nd14+nd0))
mm808.2
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avg large disk corrugation lengthLc14 ~ π / 4 * Dd14mm683.3
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avg small disk corrugation lengthLc0 ~ π / 4 * Dd0mm383.3
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avg corrugation lengthLc ~ π / 4 * Ddmm634.7
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avg num corrugations (per half disk)nc ~ (Dd/2) / (xc/2)-23.3
<-- non-integer is ok, it's averaging large and small disks
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Barrel geometry
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stave channel width (this is one side)wsmm20.65
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stave channel height at edgehs1mm2.92
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stave channel height at centerhs2mm7.49
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stave channel avg heighths = (hs1 + hs2)/2mm5.2
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stave channel cross-sectional areaAs ~ ws * hsmm²107.5
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stave channel hydraulic diameterDs = √(4 As / π)mm11.7
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variation from disk corrug hydraulic diameterDs / Dc - 1-1.6%
<-- this is pretty similar to disk corrugation hydraulic diameter
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L4 stave lengthLs4mm793
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L3 stave lengthLs3mm503
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num L4 staves (per half)ns4-35
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num L3 staves (per half)ns3-22
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avg stave channel lengthLs = (Ls4*ns4+Ls3*ns3)/(ns4+ns3)mm681.1
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variation from disk corrug avg lengthLs / Lc - 1-7.3%
<-- this is pretty similar to disk average corrugation length
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Note: given geometric similarity of average barrel staves to average disk corrugation channels, I treat them all as basically the same in flow calcs below.
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num half disks equivalent to num stavesnds_equiv = (ns4 + ns3) / nc2.4
<-- non-integer is ok, it's averaging lots of things
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Module power loads
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reference link for power loadsCopy of EIC-LAS Power - 2025-10-22
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power per RSUQ_1rsuW0.2
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power per LECQ_1lecW0.2
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power per AncASICQ_1aaW0.5
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length per RSUL_rsumm23
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num rsu per avg corrug length (exact coverage)n_rsu = Lc / L_rsu-27.6
<-- non-integer is ok, it's averaging stuff
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typical num RSU per modulen_rsu_per_mod-5.5
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total RSU power per avg corrugQc_rsu = Q_1rsu * n_rsuW5.52
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total LEC power per avg corrugQc_lec = Q_1lec * n_rsu / n_rsu_per_modW1.00
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total AncASIC power per avg corrugQc_aa = Q_1aa * n_rsu / n_rsu_per_modW2.51
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total power per avg corrugQc = Qc_rsu + Qc_lec + Qc_aaW9.03
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total RSUs modeled below
n_rsu_system = n_rsu * L₄ * N₄ / n_rsu_per_mod
-2209.7
<-- cross-check this against others' current-best-estimates
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Boundary conditions
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source temperatureTₛK283
<-- in practice, we would control a heater on the input airstream outside the detector until T₁ below is as desired
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source pressurePₛbar5.856
Iteratively solved, so always check carefully for reasonability, and that final outlet pressure below = Pₑ boundary condition.
If it misbehaves, the usual move is (1) Copy the formula. (2) Type in a numeric value that's a decent initial guess. (3) Paste the formula back in.
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""Pa593,316
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""psi86.1
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source mass flow rateṁₛkg/s1.10
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final exit pressurePₑbar1.0
atmospheric = 1.0, or possibly supplied vacuum >= ~0.3
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Each "stage" consists first of a length of pipe with a friction factor, then a change of cross-sectional area
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1 = entrance = at beginning of stage
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2 = downstream = after pipe length, but before area change
It's not trivial to add or subtract column channels in this spreadsheet.
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3 = outlet = manifold = after area change
To calculate fewer than 6 distinct channels:
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- set "nᵢ" to 1 for the dummy channel(s), i.e. no splitting of parent
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- set L, D_mm, and hs to same value as their respective parents
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stage description--source --> ePICbranches to hadron + electron sidesbranches to top + bottom halvespipes inside ePIC to half disks and half barrelsdisk channels and stavesexhaust ducts
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stage index--012345
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num channels (per parent channel)nᵢ-1227.423.30.0144
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total num channels in systemNᵢ = product(nᵢ)m1243069410
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mass flow per channelṁ = (ṁₛ or previous stage ṁ) / nᵢkg/s1.1000.5500.2750.0370.0020.110
<-- mass flow is split exactly among branches at each stage, i.e. assuming each branch has similar back-pressure
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lengthLm10.00010.00015.00010.0000.63510.000
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hydraulic diameter (mm)D_mmmm5050502311.588.274
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hydraulic diameterDm0.0500.0500.0500.0230.0120.088
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single channel cross-sectional area along length LA₁₂ = π D² / 4m²1.96E-031.96E-031.96E-034.15E-041.04E-046.12E-03
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all channels total cross-sectional areaΣA₁₂ = A₁₂,ᵢ * Nᵢm²1.96E-033.93E-037.85E-031.24E-027.22E-026.12E-02
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""cm²19.6339.2778.54123.77722.25612.00
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entrance temperatureT₁ = Tₛ or previous stage T₃K283280.5279.5279.2277.9283.6
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""°C10.07.56.56.24.910.6
<-- aim for desired temperature at corrugation or stave inlet (see highlighted cell)
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entrance pressureP₁ = Pₛ or previous stage P₃Pa593,316537,851438,530351,269134,963111,009
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entrance density (ideal gas law)ρ₁ = P₁ / (R * T₁)kg/m³7.3056.6825.4674.3841.6921.364
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entrance dynamic viscosityμ₁ = μ₀ * (T₁/T₀)^0.76Pa*s1.76E-051.75E-051.75E-051.75E-051.74E-051.77E-05
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Reynold's numberRe = 4 ṁ / (π D μ₁)-1.59E+068.00E+024.01E+021.17E+021.01E+018.98E+01