Modelling of Cryocoolers

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Brett Moody
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1 Modelling of Cryocoolers 1 Introduction Current cryocooler models in ESAAN make use of polynomial fit functions in combination with boundary nodes his limits the number of free parameters (e.g heat sink temperature) therefore Leading to a conservative design only considering worst cases It is difficult to perform sensitivities or assess the impact of sub-systems (e.g. compressor performances) A correct heat balance is not always obtained 2

2 Q input Stirling/ Pulse ube models Heat sink Compressor/warm part compressor Coldfinger g(c,qc) f(c,qin) h(c,qcool) Q cooling i(qcool,qin) Cold tip Instrument I/F tip Q input = a Q 2 cool + b Q cool + c h( c,q cool ): a = a 1 2 tip + a 2 tip + a 3 b = b 1 2 tip + b 2 tip + b 3 c = c 1 2 tip + c 2 tip + c 3 3 Q pre-cooling and Q cooling, depending on the Hx efficiency and the fluid, are provided only by polynomial fits Joule homson coolers Q input Q pre-cooling Heat sink J compressor GL (pipe, harness etc) J pre-cooling GL (pipe, harness etc) Compressor, h Intermediate Heat sink, HS Counterflow Heat exchanger Counterflow Heat exchanger Q cooling J stage J expansion, c Instrument I/F 4

3 Joule homson coolers including Physics Q input Heat sink J compressor Compressor, h Q pre cooling m H p high, high H phigh, HS H plow, high Q cooling m H p high, HS H p low, H HS p low, HS GL (pipe, harness etc) J pre-cooling GL (pipe, harness etc) J stage Intermediate Heat sink, HS Counterflow Heat exchanger Counterflow Heat exchanger J expansion, c Instrument I/F 5 Requirements for Cooler models In most cases the tip temperature is specified by the system (coming from detector needs etc) he required cooling power is obtained by the MM For sizing the thermal system the cooler model must provide Q input = f( tip, Q cooling ) For transient verification, the cooler model must be able to calculate for a given Q input. he following function is required within a MM: Q cooling = g( tip, Q input ) 6

4 Requirements for Cooler models Cooler model shall consider as a minimum the following parameters: For single stage cooler heat sink, tip, Q input, Q cooling For multistage coolers, the influence of the intermediate cooling stages for different operating conditions needs to be implemented his can not be handled by polynomial fits or would require a large amount of data points at specific conditions (e.g. isothermal points) 7 Requirements for Cooler models Overall heat balance needs to be correct Common approach is: Q dissipated = Q input For Stirling, P and reverse urbo-brayton correct approach is: Q dissipated = Q input +Q cooling Use of boundary nodes shall be limited, where required link them correctly with the MM Shall be simple, fast and robust 8

5 Stirling cooler Compressor transform electrical Energy into pneumatic Work (pv work). For high efficient space coolers one can assume: pv = Q in -I 2 R Pressure wave generated passes through a regenerative heat exchanger (= Regenerator) At the end the pressure wave and massflow wave are shifted such, that the following cooling occurs: q gross = /( - )*pv he available cooling power is: q net = q gross losses Main losses are: conductive, radiative, regenerator and shuttle losses 9 Stirling cooler losses Conductive/radiative losses are equal to parasitic loads of a non-operating cooler. Regenerator losses: Q regenerator = dm/dt * c p *( - ) (1- ) with dm/dt ~ pv/ Shuttle losses Q shuttle ~ ( - )/stroke 1

6 Stirling cooler equation It should be possible to describe the Stirling cooler with the following equation: pv c( ) a(1 reg ) pv ( ) b stroke q net Approach has been tested with MIPAS 5-8k Astrium cooler data, assuming a constant compressor efficiency (due to the lack of I 2 R data) 11 Stirling cooler extrapolation Dataset includes 18 measurements from 57-15K and -1 C to 4 C heat sink pv =.8 *Q in C = 22 mw /193K from parasitic measurements, refined during interpolation Distinguishing between regenerator and shuttle loss not possible, but regenerator loss as a function of displacer stroke )) a(1-eps)/ / (a( Re ge ne rator losse s / y = e x R 2 = Stroke 2.4 Stroke 3.4 Expon. (Stroke 2.4) Expon. (Stroke 3.4) y = e x R 2 =

7 Stirling cooler extrapolation uncertainty Strong deviations from measurements above 15K, additional correction for these temperatures required 2% 15% 1% 5% % -5% -1% -15% uncertainty in cooling power [K] 13 Stirling cooler model Cooler warm unit Result MM: Cooler function Q input q net GL=.71*1-3 GR =.176 (3mW q instr,cooler Wu 263K) = 64.9K -> 12.19W input Q in = 11.17W -> 66.1K c q instr tip Cooler function:.8 * Q input ( e ) q net Note: not valid for temperatures below 5K and above 15K 14

8 Pulse ube cooler Compressor transform electrical Energy into pneumatic Work (pv work). For high efficient space coolers one can assume: pv = Q in -I 2 R Pressure wave generated passes through a regenerative heat exchanger (= Regenerator) At the end the pressure wave and massflow wave are shifted such, that the following cooling occurs: q gross = / )*pv he available cooling power is: q net = q gross losses Main losses are: conductive, radiative, regenerator and pressure losses 15 Pulse ube cooler losses Conductive/radiative losses are equal to parasitic loads of a non-operating cooler. Regenerator losses: Q regenerator = dm/dt * c p *( - ) (1- ) with dm/dt ~ pv/ Pressure losses Q press ~ f(p ampl ) * c / h *pv p ampl ~ Q in Data taken from MPC Air liquide RP cooler 16

9 remaining/qgross Pulse ube cooler extrapolation Dataset includes 23 measurements from 4-95K and -4 C to 4 C heat sink pv =.7 *Q in C = 465 mw /28K from parasitic measurements Pressure losses )) / (a(1- y = E-3x E-2 R 2 = E-1.1 1% % 7 Qin 15% Regenerator losses y = e x R 2 = uncertainty c/h % % -1% -15% 17 Pulse ube cooler model Cooler function Q input q net Cooler warm unit tip GL= 1.77*1-3 GR=1.8*1-4 q instr Cooler function: ( ) input net.7 * Q *1 - * Q input e q Result MM: = 6.63K, =31K, q instr = Q in = 45W -> 58.7 K c = 75.43K, = 273K, q instr = 1W -> 36.3 W Q in = 35W -> 76.3 K c Note: function not valid for input powers below 3W and tip above 1K, not verified for high tip 18

10 Conclusion Performance predictions of Stirling and P coolers is feasible by interpolating the various loss mechanisms, requiring much less input data than polynomial fits Some additional effort is required for high tip temperatures, mainly for cool down predictions In addition to the classical thermal parameters, the compressor efficiency also needs to be measured 19 Future work Distinguish between Compressor and warm part of finger and predict dissipated power on each Modelling of the gas temperature as a function of gross cooling power and I/F temperature (to improve high temperature performance) Extend model to multistage Stirling and Pulse ube cooler 2

Development of a 4K Pulse-Tube Cold Finger for Space Applications

Development of a 4K Pulse-Tube Cold Finger for Space Applications A. Charrier 1, I. Charles 1, B. Rousset 1, C. Daniel 2 and J.-M. Duval 1 1 Univ. Grenoble Alpes, CEA INAC-SBT, Grenoble, France 2 CNES