ANALYTICAL MODEL FOR THE BYPASS VALVE IN A LOOP HEAT PIPE

Transcription

1 ANALYTICAL MODEL FOR THE BYPASS ALE IN A LOOP HEAT PIPE Michel Seetjens & Camilo Rindt Laboratory for Energy Technology Mechanical Engineering Deartment Eindhoven University of Technology The Netherlands December 6 MATEO-ANTASME deliverable 9.

2 Introduction The loo heat ie (LHP) is art of the cryo-cooling system for the thermal control of the Alha Magnetic Sectrometer used for extra-terrestrial studies on anti-matter. The LHP basically comrises a conventional cooling cycle (i.e. heat intake by an evaorator; heat release by a condensor) following the schematic in Figure a. The circulation of the working fluid (roylene) is set u by caillary forces in the evaorator according to the heat-ie rincile (hence the denotation LHP ). In order to avoid freezing of the working fluid and consequently cessation of the circulation and thus termination of the cooling rocess a byass valve (comonent in Figure a; cross-section in Figure b) has been laced between evaorator and condensor that interruts the circulation in case the evaorator ressure e (tyically the saturation ressure) dros below a certain lower limit min ( back ressure ). The system has two steady-state oerating modes: e > min : only liquid-vaour hase change ossible; no risk of freezing. The byass valve is closed and the circulation is enabled. The LHP is oerational. e min : solid-vaour hase change ossible; risk of freezing. The byass valve is oened and the circulation is halted; the working fluid (vaour) is redirected into the evaorator through the comensation chamber (comonent C in Figure a). The LHP is not oerational. To date a oor descrition exists of the thermodynamical behaviour of the byass valve. Better descrition is essential so as to better describe the behaviour of the LHP as a whole however. To this end the thermodynamical behaviour of the byass valve in its two steadystate oerating modes is to be erformed first through an analytical integral analysis and as a follow-u on the latter through a detailed numerical analysis using the finite-volume method (FM). The discussion hereafter concerns the analytical integral analysis. Results of the FM analysis are communicated searately. a) Schematic of the LHP. b) Cross-section of the byass valve. Figure : The Loo Heat Pie (anel a) and the byass valve (anel b). Comonents and C in anel a indicate the byass valve and comensation chamber resectively.

3 Integral thermodynamical analysis The byass valve corresonds in both oerating modes with an oen two-ort thermodynamical system i.e. having one inlet (denoted by i ) and one outlet (denoted by o ) (Shavit & Gutfinger (995)). Imortant to note is that viscous effects and inhomogeneous roerty distributions on the orts are not taken into account in the integral analysis. These issues are dealt to be with in the abovementioned FM analysis. The integral thermodynamical behaviour of the byass valve is in both oerating modes governed by the integral conservation laws for mass and energy. These laws are given by Q ṁ = i A i i = o A o u o ṁ + h i + i = h o + o () with ṁ A and h signifying mass flow density cross-sectional area velocity and secific enthaly resectively and Q reresenting heat loss by radiation. Closure of the roblem requires secification of the state of the working medium (roylene) as a function of the thermodynamical variables. Proylene may to good aroximation be considered an ideal gas (Chao & Zwolinksi (975)). This comletes the integral model with the equation of state and the enthaly relation = RT h = c (T )T c (T ) = a k T k () k= resectively with R the secific gas constant and c (T ) the temerature-deendent secific heat with coefficients a k following Çengel & Boles (). Relevant quantities are the thermodynamic variables and T and the fluid-dynamical variables and φ = A (volumetric flow rate) at the inlet and outlet of the byass valve. Given are the mass flow ṁ the radiative heat loss Q the cross-sectional areas A i and A o and the inlet conditions i T i and i. This straightforwardly leads to φ i = A i i = i RT i () and thereby fully determines the state at the inlet. The state at the outlet is determined as follows. Recasting relations ()-() yields o A o o o = ṁ = RT o c (T i )T i + Q { [ o ṁ = c (T o )T o + i ] i T o A i } () o T i A o roviding three equations for the four unknown outlet quantities o o T o and o. Imosing the ressure gradient = o i < fixes the outlet ressure o and via exressions () fully determines the state at the outlet as a function of. This is elaborated below. The state at the outlet deends on the relative share of the three energy contributions (radiative heat loss; kinetic energy; enthaly flux) in the total energy balance (nonlinear relation in ()). Consider to this end relations () in the non-dimensional form = c ( ) + { [ ΛT + ] } = + = Λ φ = (5) in terms of the non-dimensional outlet conditions = i = T o T i = o i = o i φ = φ o φ i (6)

4 with c ( ) = c (T i )/c (T i ). (Form (5) readily follows from rescaling ()). The corresonding arameters read = Q ṁc (T i )T i = radiative heat loss = enthaly flux i kinetic energy = c (T i )T i enthaly flux Λ = A i A o (7) with arameters and relating the three energy contributions as indicated; the indeendent outlet variable < controls the ressure dro over the byass valve. The ratio of cross-sectional areas is fixed at Λ = and thus = φ hereafter. Figure gives a visual reresentation of the thermodynamical behaviour of the byass valve. Shown are the deendent outlet variables and (= ) as a function of and with growing non-dimensional radiative heat losses. The grahs clearly demonstrate the changes in outlet conditions with changing and. Correlations for normal oerating conditions Tyical values for the various quantities under normal oerating conditions are (Bodendiek et al. (5)): T i 5 K i O( 5 m/s) ṁ O( 5 kg/s) Ḣ i = ṁc (T i )T i O( W ). This gives O( 5 ) and thus imlies that kinetic effects are negligible. Under this roviso relations (5) simlify to = c ( ) = + = + (8) resulting in a constant outlet temerature for given and consequently roortional and inversely-roortional deendence of and resectively uon. The non-dimensional outlet temerature is to good aroximation given by =. This admits further simlification of (5) to the ractical correlations = = + = + (9) that conveniently relate the inlet and outlet conditions of the byass valve for given. Figure gives the outlet conditions as a function of according to (8) (heavy) and aroximated by (9) (dashed) with growing non-dimensional radiative heat losses. The correlations (9) rogressively deart from relations (8) with increasing. However deviations remain sufficiently small for correlations (9) to rovide an adequate descrition of the thermodynamical behaviour of the byass valve. Conclusions and outlook The above study concerns an integral thermodynamical analysis of the byass valve in its two steady-state oerating modes. This study resulted in the ractical correlations (9) between the inlet and outlet conditions of the byass valve. Omitted in this analysis are irreversible effects occurring in the interior of the byass valve due to viscosity. The influence of such effects on the relations between the several thermodynamical quantities deends essentially on the oerating mode and is to be investigated through a detailed numerical analysis of the byass valve using the finite-volume method (FM). Results of the FM analysis are resented in a forthcoming communication.

5 for = T = for = = for = b) c) = for =. = for =. for =. T.5 a) e) f) = for =. = for =. for =. T.5 d) g) h).5 i) Figure : isual reresentation of the thermodynamical behaviour of the byass valve. Shown are the deendent outlet variables T and (= ) as a function of and with growing non-dimensional radiative heat losses. References Shavit A. & Gutfinger C. 995 Thermodynamics. From Concets to Alications Prenctice Hall London. 5

6 outlet conditions for = and = outlet conditions for =. and = a) outlet conditions for =. and = b) outlet conditions for =. and = c) d) Figure : Relations between inlet and outlet conditions of the byass valve with increasing radiative heat losses. Heavy lines corresond with relations (8); dashed lines corresond with the ractical correlations (9). Chao J. C. & Zwolinksi B. J. 975 ideal gas thermodynamic roerties of ethylene and roylene. J. Phys. Chem. Ref. Data 5 6. Çengel Y. A. & Boles M. A. Thermodynamics. An Engineering Aroach Mc- Graw Hill (fourth edition) Boston. Bodendiek F. Hollenbach B. & Goncharov K. 5 Proylene loo heat ie: technical note. Document AMS-OHB-TEN- OHB-System AG Bremen. 6