Cryogenics 85 (2017) 30–43
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Cryogenics
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Thermodynamic calculations of a two-phase thermosyphon loop for cold neutron sources ⇑ Victor-O. de Haan a, , René Gommers b, J. Michael Rowe c a BonPhysics Research and Investigations BV, Laan van Heemstede 38, 3297AJ Puttershoek, The Netherlands b Reactor Institute Delft, Delft University of Technology, Mekelweg 15, 2629JB Delft, The Netherlands c Department of Materials Science and Eng., University of Maryland, NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899-8562, United States article info abstract
Article history: A new method is described for thermodynamic calculations of a two-phase thermosyphon loop based on Received 20 January 2017 a one-dimensional finite element division, where each time-step is split up in a change of enthalpy and a Received in revised form 23 May 2017 change in entropy. The method enables the investigation of process responses for a cooling loop from Accepted 24 May 2017 room temperature down to cryogenic temperatures. The method is applied for the simulation of two dis- Available online 27 May 2017 tinct thermosyphon loops: a two-phase deuterium and a two-phase hydrogen thermosyphon loop. The simulated process responses are compared to measurements on these loops. The comparisons show that Keywords: the method can be used to optimize the design of such loops with respect to performance and resulting Thermosyphon loop void fractions. Two-phase flow Ó Numerical modeling 2017 Elsevier Ltd. All rights reserved. Hydrogen Deuterium Void fraction
1. Introduction a critical parameter and hence the actual void fraction in the moderator cell determines the performance of the neutron source. A good understanding of two-phase thermosyphon loops is A numerical model is created to study trends in the dynamics of crucial for further development and optimization of efficient the thermosyphon loop. The model is based on a linear finite ele- cryogenic cooling loops. Specifically for the development of ment division of the loop where each element has certain charac- efficient moderators for cold neutrons it is of paramount impor- teristics. The model is solved using the principles of conservation tance to understand the thermosyphon loop and its dynamic of mass, energy and momentum. characteristics. Similar models haven been constructed in literature by for Although the development of cold neutron sources already instance Dobson [3,4], Kaya [5], Bielinski [6] and Zhang [7]. The started in the 1970’s, the current development of stronger neutron main difference here is the way that the flow dynamics are incor- sources (for instance at the European Spallation Source [1]) calls porated. Basically, it comes down to acknowledging the fact that for the investigation of the efficiency of the cooling design. Also, temperature changes due to (changing) heat flows are in general much as in case of the OYSTER project of the Delft University of slower than pressure changes due to changing mass flows. Hence, the Technology [2], if optimization of the performance of both neutron acceleration term of the flowing mass is ignored and a steady flow moderation and cooling efficiency is needed, a detailed knowledge state is assumed at every time step. This is known as the quasi- of the cooling process involved is required. In case of a two-phase static motion approximation. The mass flow in and out an element thermosyphon loop, the void fraction inside the moderator cell is a is the same. However, if this is strictly followed the mass in the ele- critical factor for efficient moderation of neutrons. For a two-phase ment cannot change. To mitigate this, it is assumed that mass flow, the void consists of gaseous moderator (for instance transport from one element to another is adiabatic, where the total hydrogen or deuterium vapor) which has a much smaller density entropy of the mass is constant. than the liquid part. However, for optimal moderation, density is 2. System description
⇑ Corresponding author. The moderator cell and its content are cooled by means of a E-mail address: [email protected] (V.-O. de Haan). shell and tube heat exchanger. The moderating material itself is http://dx.doi.org/10.1016/j.cryogenics.2017.05.006 0011-2275/Ó 2017 Elsevier Ltd. All rights reserved. V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 31 used to transport the heat load from the moderator cell towards the behavior between room temperature and cryogenic tempera- the heat exchanger. The transport is provided by the ther- ture so that the transition from 1 phase into 2 phases and vice mosyphon principle as shown in Fig. 1. The heat load is due to versa should be considered. This is implemented by an appropriate the nuclear gamma and fast neutron radiation accompanying the form of the conservation laws. cold neutron radiation. Also, thermal and cold neutron capture in both moderator and moderator cell wall contribute to the heat load. 3. Method The thermosyphon is a loop containing an evaporator (modera- tor cell), a condenser (heat exchanger) and a buffer volume. The The method is based on the use of a 1D finite element method as driving force for the mass transport is gravity-based and therefore used previously by Dobson [4]. All the parts of the thermosyphon fail-safe. A buffer volume is connected to the loop for inventory loop (see Fig. 1, left side) are coded by means of 1 or more elements control and storage during the hot or non-operating mode of the that are connected to one, two or three other elements depending loop. The gaseous moderator rising in the riser is condensed in on the function of the part. For instance, the riser connecting the the tubes of the heat exchanger by means of film condensation. moderator cell with the heat exchangers is modeled by 15 ele- The condensation heat is removed by sufficient helium flow. The ments. The first element of the riser is connected to the moderator condensate is not sufficient to block the condensing tubes inner element and the last element is connected to the heat exchanger. In diameter, hence the gas pressure in heat exchanger is almost a sta- this way a loop of elements is constructed from heat exchanger via tic pressure. Further, the diameter of the downcomer should be the downcomer to the moderator cell and via the riser back to the sufficiently large to let a liquid film drop into the moderator cell. heat exchanger. The mass flow in this loop is everywhere the same. In the downcomer the film is changed into a complete filled tube, In a similar way another series of elements is constructed between from which point the pressures starts to increase due to the weight the buffer and the heat exchanger where the mass flow is also con- of the liquid column. Then, there is also some slight sub-cooling of stant, although different from the mass flow in the loop (see Fig. 1, the liquid. In the moderator cell the liquid is evaporated, taking up right side). The same is done for the parts of the helium loop. The the heat load. The vapor bubbles accelerate (almost without fric- loops are connected at the heat exchanger, where the heat load of tion) to the top of the moderator cell where they merge and enter the heat exchanger of the thermosyphon loop equals the heat sink the riser. The vapor is then raised to the top of the heat exchanger, of the heat exchanger of the helium loop. where the loop restarts. The pressure drop during the transport The mass flow through the elements results in a pressure drop through the riser is compensated by the pressure provided by because of the friction with the wall. The momentum pressure the weight of the liquid column. drop due to evaporation or condensation is neglected (see Appen- Under static conditions, the complete heat load provided to the dix F). For constant mass flows, starting from the pressure in the moderator cell (and other parts) is transported to the heat- buffer, all pressures in the elements are fixed. The simulation is exchanger and removed by the helium. Under dynamic conditions performed by defining a time step and determining for each ele- the temperature of all parts change in time and the temperature ment the heat load or sink during that time step. This heat load dependent heat capacity of all items must be taken into account. or sink changes the enthalpy of the contents resulting in a changed It is essential to treat the moderator cell differently from the pressure and temperature in an element. This change in pressure supply and return lines: it is closer to pool boiling than to normal and temperature give rise to a change in mass in the element that two-phase flow in pipes. This is treated in A. A similar argument is calculated by means of the concept of adiabatic mass transport holds for the tube side of the heat exchanger. This is much closer resulting in pressure and mass flow changes. After the new masses to plate condensing than to normal two-phase flow. This is treated and pressures have been calculated the next time step is applied. in B. Simulations of thermosyphon loops have been performed by for 3.1. Definition of an element instance Dobson [3,4], Bielinski [6] and Zhang [7]. However in these cases a limited temperature range has been applied where An element is defined as a container interacting with the envi- the loop is either always single-phase or two-phases. Here we need ronment and the material inside the container with 1–3 inputs
Fig. 1. Thermosyphon loop (left) divided in linear elements (right). 32 V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 and/or outputs. An element with 1 input/output is used to simulate W kC Q ¼ AC ðTo TiÞð3Þ the buffer. An element with 3 inputs/outputs is used to simulate dC the part of the thermosyphon loop where the buffer is connected, where k is the thermal conductivity of the wall material, and d the here the heat exchanger. C C thickness of the container wall. It might also be temperature depen- The gas and/or liquid interacts via the inputs and outputs with dent. Furthermore, the container can be subjected to an additional the other elements. The interaction is based on three conservation heat load (or sink) at the surface of the container, denoted by PS. laws: i.e. mass, energy and momentum, and the assumption of Lastly the nuclear radiation heat load is taken into account by the quasi-static motion. This means that the acceleration of the flow W is assumed to be zero and the rate at which pressure waves are container wall heat load density P and moderator heat load den- propagated through the material is much higher than the rate at sity P both expressed in W/kg. They can be calculated by multiply- which mass and heat moves in the loop. This assumption is ing the specific nuclear radiation heat load (i.e. the nuclear heat allowed as the velocity of the material is much less than the veloc- load per kg material experienced due to a reactor power of ity of sound. Hence it is safe to assume a quasi-static solution for 1 MW) by the nuclear power of the reactor, PN expressed in MW. the equation of motion, so that Bernoulli’s law (the sum of static The temperature of the container varies depending on the applied pressure, dynamic pressure and gravitational pressure is constant) loads, mass flow and material of the container. The way this is taken and the gas law for gasses (pressure as a function of temperature into account is addressed in Appendix C and Appendix D. and density) can be applied. Hence, it is assumed the dynamical The container of each element contains gas and/or liquid. The effects are due only to the temperature changes in the system temperature difference between the inside container wall and and the heat transfer. the gas and/or liquid results in heat transport between the con- Each input/output of an element is connected to 1 input/output tainer wall and the contents according to of another element, and at the connections the mass flows and Q ¼ AC hC ðTi TÞð4Þ pressures are equal.
where hC is the heat transfer coefficient between wall and contents. 3.2. Definition of a container This coefficient depends on the phase and the kind of flow. Here, for an assumed homogeneous flow inside a circular pipe, it is approxi- The energy flows towards and inside the container are shown in mated by using the Dittus-Boelter relation (as introduced by McA- Fig. 2. The container exchanges heat with the environment due to a dams [8]) between the Nusselt number, the Reynolds number and temperature difference. The heat flow from or to the environment the Prandtl number is divided into two parts: the heat flow due to conduction and the : : Nu ¼ 0:023Re0 8Pr0 4 ð5Þ one due to thermal radiation. The conduction heat is proportional to the temperature difference between container wall and environ- where Reynolds number is given by ment, hence qvD 4 U C Re ¼ ¼ ð6Þ Q ¼ AC heðTe ToÞð1Þ g p gD v where To is the outside temperature of the container wall, Te is the where q is the density, is the velocity, g is the dynamic viscosity U environmental temperature, AC is the effective area of the container and D the effective diameter of the pipe cross section and is the and he is the heat transfer coefficient of the container to the envi- mass flow. The Prandtl number is given by ronment. It is determined by the heat transfer coefficients between C g Pr ¼ p ð7Þ wall and environment and the effective areas involved. It might be k temperature dependent. The radiation heat flow is proportional to and is a material property depending on temperature. C is the heat the 4th power of the temperatures according to p capacity of the gas or liquid and k its thermal conductivity. From the R 4 4 definition of Nusselt number Q ¼ AC C rBðTe To Þð2Þ : 8 2 4 kNu where rB ¼ 5 67 10 W/(m K ) is Stephan-Boltzmanns constant h ¼ : ð8Þ C D and C is the emissivity coefficient. When super insulation is applied this might be different as this reduces radiation by a factor of two An exception for this is the heat transfer coefficient in the heat per layer. Here this is incorporated by reducing the emissivity coef- exchangers where the gas condenses on the wall into a thin film ficient. The conduction heat flow from the outside to the inside of which starts flowing down. The way the heat transfer coefficient the container wall is determined by is calculated is explained in Appendix B.
3.3. Definition of an input/output
Each input/output is characterized by 5 parameters: Gravitational height. As gravitation is the driving force for the thermosyphon loop, each input/output is characterized by a speci- fic height defining the height difference between the input/output and the center of mass of the element. This is referred to as gravi- tational height. The pressure difference due to the gravitational height between the input and output of an element must be com- pensated by a difference in pressure between two consecutive ele- ments or an internal resistance so that the resulting flow is stationary in order to conform to the principle of the quasi-static solution. Fig. 2. Overview of energy flows in container wall. From left to right: container (red), wall (blue) and environment (green). (For interpretation of the references to Liquid and Gaseous mass flow. As a two phase flow is considered, colour in this figure legend, the reader is referred to the web version of this article.) both the liquid and gaseous (or vapor) flow at each input should be V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 33 defined. In such a case the total mass flow and total enthalpy flow is known. Pressure at input/output. The pressure of the liquid and gas are assumed to be identical at any point in the loop. This corresponds to the quasi-static motion approximation. The pressure at an input/ output of an element will be set equal to the pressure at an input/ output of a connected element. Friction coefficient The friction coefficient defines the relation between the pressure drop in an element from one input/output to another one and the mass flowing between these parts. In gen- eral it is dependent on the density of the material and the type of flow. Here three possibilities are considered. Pure vapor flow, pure liquid flow and two-phase flow. The way the pressure drop (i.e. the effective friction coefficient) is calculated is explained in Appendix F.
3.4. Material properties
For the calculations the material properties over the relevant temperature range are needed. For the container wall two materi- als are considered: aluminum and stainless steel. Hydrogen properties are taken from the equation of state pub- lished by Leachman [9] for normal hydrogen. Hydrogen molecules can have two different energy states, depending on the mutual spin direction of the constituting atoms. For ortho-hydrogen the spins are parallel, for para-hydrogen the spins are anti-parallel. Normal hydrogen is hydrogen for which the ortho- and para-hydrogen molecules are in thermal equilibrium. In the calculations, the dif- ferences in material properties due to the transitions between ortho and para hydrogen are ignored. Using the equation of state the enthalpy, entropy and pressure can be calculated as function of temperature and density. The density for the vapor and liquid is calculated as a function of the saturation pressure at a certain temperature. The thermal conductivity of hydrogen is taken from Assael [10] and its dynamic viscosity from Muzny [11]. Deuterium properties are taken from the equation of state pub- lished by Richardson [12] for normal deuterium. The differences in material properties due to the transitions between ortho and para deuterium are ignored. The thermal conductivity and dynamic vis- cosity of deuterium are taken from the NIST Standard Reference Database [13]. Helium properties are taken from Bell [14]. For alu- minum the properties of alloy Al6061 are used and for stainless steel the properties of alloy SS304 as published by Duthil [15].
3.5. Iterations Fig. 3. Calculation scheme of thermosyphon loop. The dynamic behavior is calculated by iterative time step. The iteration steps are schematically shown in Fig. 3.
During the time between each time step j (from time tj to tjþ1)it 8 9 > L V > is assumed that all time-derivatives are constant, like mass flow > P ; ðm þ m ÞþQ ; ðt t Þþ > <> k j k;j k;j k j jþ1 j => and heat flow. For each element k the conservation laws are P n L L V V 8k k U H þ U H ðt t Þ¼ ð10Þ applied to determine the starting values for the next time step. > i¼1 i;k;j i;k;j i;k;j i;k;j jþ1 j > > > Conservation of mass. The total mass in the loop is constant, hence : L L V V L L V V ; m ; H ; þ m ; H ; m ; H ; m ; H ; UL k jþ1 k jþ1 k jþ1 k jþ1 k j k j k j k j the total mass flowing in or out an element (liquid mass flow i;k;j V L U where H ; ; is the enthalpy of the liquid at element k and time step j and gaseous mass flow i;k;j) is equal to the mass accumulation or i k j UL < L reduction if i;k;j 0 and otherwise Hi;k;j is the enthalpy of the liquid at the ()element connected to input/ouput i. It is determined by the local Xnk UL UV L V L V V 8k i;k;j þ i;k;j ðtjþ1 tjÞ¼mk;jþ1 þ mk;jþ1 mk;j mk;j ð9Þ density and temperature of the liquid. Hi;k;j is the enthalpy of vapor i¼1 UV < V (or gas) at element k and time step j if i;k;j 0 and otherwise Hi;k;j is where k ranges over all elements, i over all inputs/outputs and j is the enthalpy of the vapor at the element connected to input/ouput i. the time step considered. It is determined by the local density and temperature of the vapor Conservation of energy. The change of energy in the content (or gas). mass is equal to the energy gain/loss from/towards the wall or The change of energy of the container wall is equal to the by the flow of mass, hence energy gain/loss from/towards the environment, hence 34 V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 () Xnk means of Eq. (11). Details of the calculation can be found in Appen- W S C C C : 8k Pk;j þ Q k;j Q k;j ðtjþ1 tjÞ¼mk ðHk;jþ1 Hk;jÞ ð11Þ dix C. The contents response might be translated into a pressure i¼1 and temperature transient or into evaporation or condensation C changing the void fraction of the element. This depends on the where Hk;j is the enthalpy of container material at element k and state of the content in the element and the interaction with the time step j. It is determined by the local density and temperature container wall. By means of Eqs. (9) and (10) it is possible to calcu- of the container. Q S PS Q R Q C is the energy flow from the ¼ þ þ late the values of the pressure and temperature of the liquid and environment to the container wall. vapor (or gas) mass. Details of these calculations are described in Conservation of momentum. The conservation of momentum in Appendix D. Finally, the mass distribution at the end of the step the loop is used to determine the pressure changes in the loop. is calculated. As argued the quasi-static motion approximation is The pressure difference between element and input/output of an used so there is no net mass flow in or out an element. However, element is given by if this is strictly followed, unnatural behavior occurs as the masses () D U2 in the element could not change. Hence, it is assumed that adia- i Li;k i;k;j ; D ; 8k i pk;j ðpk;j qk;jg hi kÞ ¼ f i;k;j 2 ð12Þ batic mass transport occurs where the total entropy of the mass 2q ; DAC k j will remain constant. This is described in Appendix E. The third step is to calculate the flow conditions due to the con- where g is the gravitation constant and DL ; is the effective length of i k tent state in each element by using Eqs. (12) and (13). the input/output to the element. The effective friction factors f , i;k;j It should be noted that for Eq. (10) also the enthalpy at the end mass flows U ; ; and densities are calculated according to the i k j qk;j of the time step is needed. The enthalpy depends on the pressure method described in Appendix F. and temperature that will change between steps. Hence, the above ; Each input/output of an element k i is connected to 1 input/out- steps 2 and 3 need to be repeated until no relevant changes occur. ; put of another element k i, at the connection the mass flows and If this procedure does not converge, the time step must be reduced pressures are equal: until it does.
i i0 pk;j ¼ pk0;j UL UL : 4. Simulations and results i;k;j ¼ i0;k0;j ð13Þ UV UV i;k;j ¼ i0;k0;j As an example two thermosyphon loops are considered. The first example is the thermosyphon loop used to cool down a hydro- Initial state. Initially the loop is filled with a certain amount of gen moderator at NIST. The second example is from a publication gas at room temperature. All the mass flows are zero and the loop in which a deuterium thermosyphon loop is described. is in pressure equilibrium including the difference in hydrostatic pressure. All heat loads/sinks are initially zero and to start the cal- culation, a change in heat load or heat sink is applied at some ele- 4.1. Example 1 ment(s) of the loop. The first calculation step is to determine the quantity of these NIST PeeWee [16] hydrogen cold neutron source is simulated. heat loads (or heat sinks) that might depend on the density, tem- The thermosyphon loop of PeeWee, as shown in Fig. 4, is cooled perature or dimensions of the materials in the loop. by means of a plate-and-fin heat exchanger. This was simulated The second step is to determine the container wall and contents by a tube-and-shell heat exchanger with similar thermal transport response to this heat load. The container wall can be calculated by properties and pressure drop values. The design data used for the
Fig. 4. NIST PeeWee hydrogen cold neutron source thermosyphon loop. V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 35
Table 1 Design data PeeWee hydrogen loop.
Name Buffer Line Riser HX Down comer Cell Unit Material SS304 SS304 Al6061 Al6061 Al6061 Al6061 –
C 0.1 0.1 0.1 0.1 0.1 0.1 – 2 he 1000 1 0.001 0.001 0.001 0.1 W/m /K dW 10 2 0.78 6.6 0.78 1.38 mm Te 300 300 300 300 300 300 K P=PN 0 0 0 0 0 40 ppm/kg PW =PN 0 0 0 0 0 110 ppm/kg D 20 22 14 20 20 7.75 mm 10 10 1 1 1 1 ppm DH 0 2.5 2.43 0.27 2.255 0.09455 m DL 1 16.9 5.39 0.27 6.058 0.11 m 2 AC 1 0.041 m 3 VC 0.5 0.000436 m 3 VO 0.15 0 dm rO 0.1 2
Table 2 Design data PeeWee helium Loop.
Name Buffer Line Supply HX Retour Cooler Unit Material SS304 SS304 SS304 Al6061 SS304 Al6061 –
C 0.1 0.1 0.05 0.1 0.05 0.1 – 2 he 1000 1 0.001 0.001 0.001 0.001 W/m /K dW 0.01 0.002 0.002 0.002 0.002 0.002 m Te 300 300 300 300 300 300 K D 0.02 0.0355 0.06 0.0062 0.04 0.01 m 10 10 10 10 10 10 ppm DH 05 0.27 0.27 0 0 m DL 1 10 27 0.27 27 0.5 m 2 AC 1 1m 3 VC 0.92 0.15 m
Table 3 Volumes and masses PeeWee Loops.
Name Volume Area Content Mass Empty mass dm3 m2 gram kg Moderator Cell 0.436 0.041 8.31 0.453 Heat exchanger 6.09 5.7 82.6 14.58 CNSBuffer 500 1 81.3 80 Cold CNS Loop 7.65 6.13 125 15.6 Total CNS Loop 514.1 8.30 207 114 Cooler 150 1 1426 5.43 HXShellSide 3 2.52 21.4 0.68 CoolerBuffer 920 1 490 80 Cold Cooler Loop 263.3 12.01 2356 142 Total Cooler Loop 1293 14.12 2852 240
hydrogen loop is shown in Table 1. The helium loop is approxi- The temperature difference over the helium side of the heat mated and has been designed to give results corresponding to exchanger during cool down behaves similar and is of the same the refrigerator of the helium loop of PeeWee. The design data used order of magnitude. The temperature of the hydrogen riser close for the helium loop is shown in Table 2. Both loops are cooled to the heat exchanger behaves in a similar manner, but the temper- down starting from environmental temperature (300 K). The ature in the downcomer close to the heat exchanger behaves dif- hydrogen loop starting pressure is 500 kPa, the helium loop start- ferently (the peak at the moment the hydrogen condenses is not ing pressure is 1500 kPa. The resulting volumes and masses after reproduced). This is probably due to thermal conduction effects cooling down to approximately 23 K are shown in Table 3. The in the wall material along the flow direction that are not taken into pressure in the hydrogen loop reduced to approximately 200 kPa. account in the simulation. These are especially important as long The cool down curves shown in the left side of Fig. 5 are typical as the hydrogen flow is very small. measured values as obtained by NIST and the ones to the right side The pressure change in the hydrogen loop behaves similarly. are calculated by means of the method described above. Although First a gradual decrease until the hydrogen starts to flow and the an accurate match has not been obtained, one can conclude that thermospyphon starts its operation in a single (gas) phase. Then, time constants are of the same order of magnitude. It should be a rise in pressure occurs because the hydrogen is heated by the noted that the helium loop is approximated and hence the cool parts of the loop that are still warm. After sufficient time has down of the helium is at best a rough estimate. As soon as the passed, the loop has cooled down sufficiently so that the hydrogen hydrogen loop starts to flow the simulated temperature changes that enters the heat exchanger is condensed and the single-phase are more comparable to the actual ones. thermosyphon starts to change in a two-phase thermosyphon. This 36 V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43
Fig. 5. Cool down curves. Left side: Typical measured values as obtained by NIST. Right side: Values obtained from simulation. Top: Temperature at several locations in helium and hydrogen loop. Bottom: Hydrogen pressure as function of time during cool down.
Table 4 Design data ILL Deuterium Mockup Loop.
Name Buffer Line Riser HX Down comer Cell Unit Material SS304 SS304 Al6061 Al6061 Al6061 Al6061 –
C 0.1 0.1 0.1 0.1 0.1 0.1 – 2 he 1000 1 0.001 0.001 0.001 0.1 W/m /K dW 10 2 1 6.6 1 1 mm Te 300 300 300 300 300 300 K P=PN 0 0 0 0 0 60 ppm/kg PW =PN 0 0 0 0 0 165 ppm/kg D 20 22 24 20 20 15.4 mm 10 10 1 1 1 1 ppm DH 0 2.5 4.25 1 3.45 0.2 m DL 1 16.9 8 1 8.41 0.3 m 2 AC 8 0.132 m 3 VC 4 0.007 m 3 VO 1 0.0045 dm rO 11 increases the effective heat transfer coefficient of the heat exchan- void fraction of the moderator cell. The pressure drop due to the ger and the cooling down is accelerated and the pressure decreases flow resistance in the loop is shown in Fig. 6. The background of again until the pressure control regulates the pressure to 200 kPa. the figure is taken from Hoffmann [17] so that simulated data can directly be compared to the measurements reported. The black 4.2. Example 2 dots and hatched areas represent the measurements and estimated accuracy region of pressure drops in the loop. The black lines rep- The ILL deuterium Mockup loop as described by Hoffmann [17] resent calculations as presented by Hoffmann. The colored lines is simulated. The thermosyphon loop is cooled by means of a heat represents the same pressure drops calculated using the simula- exchanger, simulated as a tube-and-shell heat exchanger. The right tions as presented here. side of Fig. 1 is a schematic diagram of the loops and the names of The simulations and measurements give the same general the parts. The design data used for the deuterium loop is shown in trends. With increasing heating power the mass flow increases Table 4. The helium loop is approximated and has been designed to because the amount of void fraction that needs to be transported give results corresponding to the refrigerator of the helium loop of form moderator cell to heat exchanger increases. With increasing the ILL mockup. The design data used for the helium loop is shown mass flow the pressure drop increases. The pressure drop for the in Table 5. Both loops are cooled down starting from environmen- single-phase flow (downcomer) reaches a maximum and then tal temperature (300 K). The deuterium loop starting pressure is reduces again. This maximum is obtained at the maximum mass 500 kPa, the helium loop starting pressure is 1500 kPa. The result- flow (right side of Fig. 6). The mass flow reduces again as the resis- ing volumes and masses after cooling down the loops to a temper- tance of the two-phase flow in the riser increases with increasing ature of roughly 25.2 K are shown in Table 6. The pressure in the void fraction. deuterium loop reduced to approximately 150 kPa. The simulations presented here used the Müller-Steinhagen and The flow resistance of the moderator cell was approximately Heck correlation function to determine the pressure drop of the matched to the measurements and the overflow volume of the flow. This correlation function produces the smallest pressure drop moderator cell was approximately matched to get a corresponding for two-phase flow. If a different correlation function is used, the V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 37
Table 5 Design data ILL Helium Loop.
Name Buffer Line Supply HX Retour Cooler Unit Material SS304 SS304 SS304 Al6061 SS304 Al6061 –
C 0.1 0.1 0.05 0.1 0.05 0.1 – 2 he 1000 1 0.001 0.001 0.001 0.1 W/m /K dW 122 2 2 2 mm Te 300 300 300 300 300 300 K D 20 35.5 60 20 60 20 mm 10 10 10 10 10 10 ppm DH 05 11 0 0 m DL 1 10 27 1 27 0.5 m 2 AC 1 1m 3 VC 0.92 0.15 m
Table 6 Volumes and masses ILL Mockup Loops.
Name Volume Area Content mass Empty mass dm3 m2 gram kg Moderator Cell 7 0.132 1050 0.358 Heat exchanger 19 30.2 639 54 CNSBuffer 4000 8 969 640 Cold CNS Loop 32.65 31.4 2281 57.5 Total CNS Loop 4039 40.6 3252 716 Cooler 150 1 1294.6 5.4 HXShellSide 6 30.2 38.9 8.2 CoolerBuffer 920 1 473.0 80 Cold Cooler Loop 308.7 41.34 2428 176 Total Cooler Loop 1239 43.45 2960 274
Fig. 6. Flow pressure drops (left side) and mass flow (right side) as function of applied heat load for ILL deuterium Mockup loop. The background of the graph was taken from Hoffmann [17]. The black dots and hatched areas represent the measurements and estimated accuracy region of pressure drops in the loop. The black lines represent calculations as presented by Hoffmann. The colored lines represents the same pressure drops calculated using the simulations as presented here. Lower (green) line: pressure drop over downcomer, middle (blue) line: pressure drop over downcomer and moderator cell, upper (red) line: pressure drop over downcomer, moderator cell and riser. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
maximum flow shifts to smaller heating powers. The differences between Müler-Steinhagen-Heck and Friedel correlations are small (less than 15%) and the differences between Chisholm and Lockhart-Martinelli correlations are also small (less than 20%), but the latter pair deviate more from the experimental values than the former pair. Fig. 7 shows the corresponding development of the void frac- tion in the moderator cell. This curve strongly depends on initial void fraction and other parameters. The slope is 7.8%/kW. It is pos- sible to increase the void fraction entering the downcomer from 0 (single phase as in the above calculations) to 0:15 ...0:5 (the max- imum depending on the liquid contents of the heat exchanger, hence of the applied heat load). Because of the higher void fraction in the downcomer the pressure head (determined by the difference in weight of the riser and downcomer) is reduced and the mass flow is reduced so that the total pressure drop due to flow friction Fig. 7. Void fraction as function of applied heat load for ILL deuterium Mockup loop. 38 V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 is less. The behavior of the loop is strongly dependent on the actual sate film (thickness d) that starts flowing downwards due to grav- void fraction entering the downcomer. The agreement with the ity (gravity constant g). During its fall downwards more vapor is measurement is worse than in the previous case, and the maxi- condensed so that the film thickness increases. Hence the film mum in mass flow occurs for much larger heating powers. The thickness and the heat transfer coefficient changes. According to resulting void fraction in the moderator cell is significantly higher, Nusselt [18] the film thickness as function of the film length, z is but less sensitive to the actual heat load on the moderator cell (the given by slope reduces from 7.8%/kW to 5.8%/kW). "#1 4zðT T Þ kLgL 4 dðzÞ¼ i ðB:1Þ 5. Conclusions g qLðqL qV ÞH
With the method described here it is possible to simulate a and the heat transfer coefficient by thermosyphon loop and determine the order of magnitude of the "#1 L 3 L L V 4 relevant time constants and the typical behavior of the flow. The g ðk Þ q ðq q ÞH : hðzÞ¼ L ðB 2Þ method is successfully applied to two cryogenic thermosyphon 4zðT TiÞ g loops in use for the cooling of cold neutron sources. It can be used to study the void fraction in the moderator cell and yields valuable where the material properties are taken for the average film temper- = ; L information on the performance of a particular design. Accurate ature that can best be taken as ð2Ti þ TÞ 3 k is the heat conductivity simulations can be done if more data about the process conditions of the liquid at this temperature, qL is the density of the liquid, qV the (like temperature, pressure, flow rate) for changing experimental density of the vapor, gL is the dynamic viscosity of the liquid and H conditions (like for instance total mass of the loop, length and the heat released during condensation. In principle these equations diameter of pipes) are available. Then also the influence of chang- only hold for small film Reynolds numbers (<30) defined as ing design parameters can be studied. 4qLvðzÞdðzÞ Re ðzÞ¼ ðB:3Þ f gL Appendix A. Void fraction moderator cell where vðzÞ is the average velocity of the liquid in the film. However, Void fraction is defined as the relative amount of gas inside the when the Reynold number increases the heat transfer coefficient is liquid moderator. The gas is created by boiling of the liquid moder- in general higher, so that this value is conservative for calculating ator. The boiling is due to the heat load on the moderator liquid the temperature difference [4]. When averaged over the length, L and moderator containment. The heat load is partly produced by of the tubes, the heat transfer coefficient is given by the nuclear heating due to neutron and gamma absorption, partly 4 by heat radiation and partly by conduction. h ðLÞ¼ hðLÞ: ðB:4Þ a 3 Due to its shape, the moderator cell cross section from bottom to top is first increased and then decreased. This means that the Now the heat transfer due to condensation can be expressed as velocity of the flow first decreases and then increases again. The Q ¼ nh pD LðT T ÞðB:5Þ complete flow pattern is complicated and different when details a i i in the moderator cell change. Hence, it is impossible to accurately where n is the number of tubes and Di their inner diameter. determine the void fraction of the moderator cell with this method. Due to its shape, the complete flow pattern in the heat exchan- The liquid will enter the cell at the bottom and exit at the top ger is complicated and different when details change. Hence, it is (as is sensible for a boiling source). Almost no liquid will go up impossible to determine the void fraction of the heat exchanger the riser until the level of the liquid plus bubbles rising from the with this method accurately. The condensed liquid will flow in hot surfaces of the moderator cell is such that the space above the volume at the bottom of the heat exchanger and almost no liq- the liquid is a froth, at which point the froth will exit the vessel uid will go down the downcomer until the level of the liquid is with entrained liquid. This is in fact obvious from a consideration such that it can spill into the downcomer. The spill will eventually of the flow and boiling process. There is no mechanism to entrain give rise to a vapor quality of 0 of the flow into the downcomer any significant amount of liquid in the flow. The actual volumetric when the spill is large enough and the entrance of the downcomer vapor flow is low (heat input less than the critical boiling limit), so has been designed properly. we have pool boiling. It is only when the cell filling (by the total Here, the condition that no liquid leaves the heat exchanger volume of liquid plus bubbles) results in an overflow condition that until the total volume of liquid in the heat exchanger has reached the liquid is swept into the riser, where the velocity is high enough a certain volume is imposed. When this volume this is reached the to support liquid transport. vapor quality exiting the heat exchanger is taken in relation to the Here, the condition that no liquid leaves the cell until the total remaining spill volume according to a cosine profile (starting with volume of liquid in the cell has reached a certain volume is 0 and ending with 1 when the remaining spill volume is com- imposed. When the amount of liquid becomes larger, the vapor pletely filled). The volume when the vapor quality in the down- quality exiting the cell is taken in relation to the remaining volume comer starts to decrease from 1 is called the overflow volume, V O according to a cosine profile (starting with 1 and ending with 0 and must be determined by means of CFD simulations using the when the cell is completely filled). The volume when the vapor geometry of the heat exchanger or by actual measurement. The quality in the riser starts to decrease from 1 is called the overflow same holds for the remaining spill volume (determined by a factor volume, V O and can be estimated by means of pool boiling calcula- rO times the overflow volume) after which the vapor quality is 0 tions, by CFD simulations using the geometry of the cell, or by and the downcomer is completely filled with liquid. actual measurement. Appendix C. Heat conduction through container wall Appendix B. Hydrogen condensation in heat exchanger The heat released to the container walls by condensation, con- Inside the tubes vapor (temperature T) condenses releasing its vection or radiation is transported to the cold side of the container heat to the inner tube wall (temperature Ti) and forms a conden- walls by means of conduction and is used to heat up or cool down V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 39 the container wall. The temperature, T in the wall as function of lc B ¼ Tð0; 0Þ To; A ¼ B; location, x and time, t can be found by solving Fouriers Law j
2 = @T @ T where le ¼ dhe k is the Biot number for the heat resistance at the ¼ a ðC:1Þ @t @x2 outside wall. A low Biot number means that the heat resistance is mainly at the outside of the wall and not due to conduction in = where a ¼ k qCp is the thermal diffusivity, with k the thermal con- the wall. Here the Biot number at the outer side of the wall is ductivity of the wall material, q its density and Cp its heat capacity. = always small. lc ¼ dhC k is the Biot number for the heat resistance The general solution of the above equation is at the inner side of the wall. This depends on the flow rate, temper- Z no jx jx 2 ature and density. j should be a solution of Tðx; tÞ¼ AðjÞ sin þ BðjÞ cos e tj =sdj ðC:2Þ d d jðl þ l Þ tan j ¼ e c : ðC:13Þ 2 2 where s ¼ d =a. For a metal wall of 1 mm thickness this time con- j lelc stant is of the order of 1 s. A specific solution for long times is given The smallest possible absolute value of j should be used. Hence pffiffiffiffiffiffiffiffiffiffiffi by > . The solution can be rewritten as no j lelc x jx jx tj2=s Tðx; tÞ¼To þ T1 þ A sin þ B cos e : ðC:3Þ x d d d Tðx; tÞ¼To þ T1 d no For all t, for x ¼ 0 (inner side of the wall) the heat flux is given by lc jx jx tj2=s þ ðÞTð0; 0Þ To sin þ cos e : ðC:14Þ @ j d d Q S W T : þ P þ P dq ¼ k@ ðC 4Þ AC x x¼0 Appendix D. Temperature increase of the container contents where
Q D.1. All vapor content ¼ hC ðÞTð0; tÞ T ðC:5Þ AC is the heat flux towards the content due to the temperature differ- The heat released from the container walls, Q, the nuclear heat- ing of the contents, P and the mass flow through the container will ence between contents and wall, PS is the additional heat load (or change the temperature according to sink when <0) on the inner side of the wall, PW is the nuclear heat @ load density on the container wall. Hence, it is assumed that nuclear T : mCp ¼ Q þ Q f þ mP ðD 1Þ heating on the wall and the additional heat load act as constant @t fluxes from the inner side of the wall. This is not completely true, where m is the mass of the contents, Cp is the heat capacity at con- but because of the high conductivity of the wall this is a valid stant pressure, T is the temperature of the contents and t represents approximation. time. As before Q ¼ AC hC ðTi TÞ, where AC is the container surface For all t, for x ¼ d (outer side of the wall) the heat flux is given area and hC the heat transfer coefficient between container wall by (at temperature T ) and content. Q occurs because the enthalpy i f R C Q þ Q @T of the mass flow entering the container is different from the ¼ k : ðC:6Þ A @x enthalpy leaving the container, according to C x¼d UL L UV V UL L UV V Further, Q f ¼ eHe þ e He H þ H C ; : Q ¼ AC heðÞTe Tðd tÞ ðC 7Þ UL L where e is the liquid mass flow and He the enthalpy entering the ; UV V where Tðd tÞ is the outside temperature of the container wall, Te is container from the connected element and e and He the same the environmental temperature, AC is the effective area of the con- for the gaseous or vapor phase. UL is the liquid mass flow and HL tainer and h is the heat transfer coefficient of the container to the V V e the enthalpy leaving the container and U and H the same for environment and the radiation heat flow is proportional to the 4th the gaseous or vapor phase. The above Eq. (D.1) can be rewritten as power of the temperatures @y P y 4 : Q R ¼ A T4 Tðd; tÞ ðC:8Þ ¼ ðD 2Þ C C rB e fg @t Cp s
8 2 4 = = where rB ¼ 5:67 10 W/(m K ) is Stephan-Boltzmanns constant where y ¼ T Ti Q f ðAC hC Þ and s ¼ mCp ðAC hC Þ. The general solu- and C is the emissivity coefficient. This can be written as tion is R ÀÁ Q ¼ A h ðÞT Tðd; tÞ ðC:9Þ Q f þ mP = C R e TðtÞ¼Tð0Þþ T þ Tð0Þ 1 e t s : ðD:3Þ i A h where C C 2 2 This solution will be followed as long as the temperature is above hR ¼ C rB T þ Tðd; tÞ ðÞTe þ Tðd; tÞ ðC:10Þ e the saturation temperature, TS of the contents at the specific density or pressure. In case the temperature would drop below this temper- so that = < ature, (if Ti þðQ f þ mPÞ ðAC hC Þ TS) the gas starts condensing at Q R þ Q C @T the saturation temperature. In that case first the time is calculated h h T T d; t k : C:11 ¼ð R þ C ÞðÞ¼e ð Þ @ ð Þ at which condensation will start AC x x¼d T 0 T Q mP = A h The solution reads ð Þ i ð f þ Þ ð C C Þ : ts ¼ s ln = ðD 4Þ S W TS Ti ðQ þ mPÞ ðAC hC Þ P þP dq f Te T þ hC 1 T1 ¼ ; To ¼ Te T1 1 þ ; ðC:12Þ leþlc and during the remaining time, t ts, the gas will condense into liq- 1 þ l l le e c uid according to the next paragraph. 40 V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43
D.2. Vapor and liquid content Appendix E. Adiabatic mass transport to obtain new pressure equilibrium The heat released from the container walls, the nuclear heating of the contents and the mass flow through the container will The change in enthalpy during heating/cooling of the content change the vapor/liquid content according to will give rise to changing material properties, giving rise to a ÀÁchanging pressure distribution in the loop. To find the new pres- L L L V : ðm ðtÞ m ð0ÞÞðH H Þ¼ AC hC ðTi TÞþQ f þ mP t ðD 5Þ sure distribution, the pressure in the heat exchanger (or buffer) is first adapted and further the pressures in the complete loop, where m is the total mass of the content, mLðtÞ is the liquid mass at L V based on the required pressure differences due to the resistance time t. H H is the enthalpy difference between the liquid and of the mass flow. It is assumed that this distribution has occurred vapor state of the contents at content temperature, T. AC the con- during the preceding time step and that it occurred adiabatically. tainer surface area, hC the heat transfer coefficient between con- The total mass in the system is constant so that one must keep tainer wall (at temperature Ti) and content, P the nuclear heating track of where the mass of the content is going to and what energy of the contents. Note that the total mass of the content is fixed transfer is related to that. This is done for each element. L V (m ðtÞþm ðtÞ¼m) and the heat load/sink is absorbed/released by Two situations can occur. First, if the final mass density, q in an evaporation/condensation of part of this mass. element is less than its start mass density, qo then the entropy This solution will be followed as long as there is a mixture of change will be 0. This means that all mass leaving the content liquid and vapor. If the right hand side of the above equation is leaves with its start entropy density, So and for the rest of the con- positive evaporation occurs, if it is negative condensation occurs. tents no time is left to take up energy from surroundings. Second, if For condensation, the maximum time until single phase occurs is the final mass density in an element is larger than its start mass L when m ðtÞ¼m density the entropy change of the initial mass will be 0, but the entropy density, S of the additional mass is added. This means that mV ð0ÞðHL HV Þ e t ¼ ðD:6Þ the total entropy of all material is constant, as it must for ideal adi- m A h T T Q mP C C ðÞþi f þ abatic expansion/compression. so that Hence, the total relevant entropy at the start is q S þðq q ÞS uðq q ÞðE:1Þ t t o o o e o mLðtÞ¼mLð0Þ 1 þ m ; ðD:7Þ tm tm where uðxÞ is the Heaviside step function (uðxÞ¼0 for x < 0 and P uðxÞ¼1 for x 0). The first term is the total entropy in the content, t the second term is the entropy of the mass entering the content. The mV ðtÞ¼mV ð0Þ 1 ðD:8Þ tm total relevant entropy at the end is qSðq; TÞþðq qÞS uðq qÞ: ðE:2Þ or when mLðtÞ¼0 o o o The first term is the total entropy in the content, the second term is mLð0ÞðHL HV Þ tm ¼ ðD:9Þ the entropy of the mass exiting the content. The change between AC hC ðÞþTi T Q f þ mP start and end is adiabatic, hence the entropy does not change so so that that qðÞ¼ðSðq; TÞ S q q ÞðS S Þuðq q Þ: ðE:3Þ t o o e o o mLðtÞ¼mLð0Þ 1 ; ðD:10Þ tm E.1. Single phase t t mV ðtÞ¼mV ð0Þ 1 þ m ðD:11Þ tm tm For a single phase the entropy is a continuous function of the density and temperature. However, as the entropy change is real- and during the remaining time, t tm , the gas will heat up accord- ized at a final pressure,p the density and temperature are related ing to the previous paragraph or the liquid will cool down according to each other. Hence, the above equation has 1 unknown only to the next paragraph. and can be solved iteratively. In such a case the final density can only be larger than the initial density, if the final pressure is larger
D.3. All liquid content than the initial pressure, then uðq qoÞ¼1 and S S The heat released from the container walls, the nuclear heating q ¼ q o e : ðE:4Þ o Sðq; Tðq; pÞÞ S of the contents and the mass transport into the content will change e the temperature similar to the first paragraph with the same gen- In case the final pressure, p is smaller than the initial pressure, then eral solution (D.3). This solution will be followed as long as the the final density is smaller than the initial density so that temperature is below the saturation temperature, T of the con- S uðq qoÞ¼0 and tents at the specific density or pressure. In case the temperature ; ; : : = > Sðq Tðq pÞÞ ¼ So ðE 5Þ would rise above this temperature, (if Ti þðQ f þ mPÞ ðAC hC Þ TS) the liquid starts evaporating at the saturation temperature. In that case, first the time at which evaporation will start is calculated E.2. Two phases according to Two phases can only occur when the temperature is at the sat- Ti Tð0ÞþðQ þ mPÞ=ðAC hC Þ t ¼ s ln f ðD:12Þ uration temperature at the final pressure, p. Then the entropy den- s T T þðQ þ mPÞ=ðA h Þ i S f C C sity of the liquid phase, SL and the entropy density of the vapor V and during the remaining time, t ts, the gas will evaporate accord- phase, S are fixed as are the densities of the liquid, qL and vapor ing to the previous paragraph. qV phase. In such a case a change in entropy is realized by either V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43 41
condensation or evaporation. The average density of the content is section. /V is the mass flow. The material properties depend on defined as the temperature and pressure of the vapor. For smooth pipes the : friction coefficient reduces to q ¼ aqV þð1 aÞqL ðE 6Þ 64 where aq is the mass per volume of the vapor part and ð1 aÞq is f ¼ ðF:4Þ V L Re the mass per volume of the liquid part. The vapor quality, x is defined as the ratio between the mass of the vapor part and the if Re < 2100; otherwise: total mass, hence 2 f ¼ fg0:79 ln Re 1:64 : ðF:5Þ aqV qL q qV x ¼ ¼ : ðE:7Þ The pressure difference per element length between element and q q q q L V input/output is given by The average entropy density can be written as D v2 2 p qV V /V V L ¼ f ¼ f : ðF:6Þ S ¼ xS þð1 xÞS ðE:8Þ DL 2D 2 2DA qV so that the total entropy of the content at the end is c o : F.2. Pure liquid flow qS ¼ qS þ qoS ðE 9Þ with For pure liquid flow the vapor flow is absent and the total flow q SL q SV q q is considered single phase flow. In such a case the friction coeffi- Sc L V and So L V SV SL : E:10 ¼ ¼ ð Þ ð Þ cient for a pipe is given by the Darcy friction factor and the pres- qL qV qoðqL qV Þ sure difference per element length between element and input/ c o Note that S and S are material properties that only depend on the output is given by the above Eq. (F.6), where instead of the vapor required pressure after the step. The change between start and end properties the liquid properties are taken. is adiabatic hence the entropy does not change : F.3. Two-phase flow qS ¼ qSo þðq qoÞðSe SoÞuðq qoÞðE 11Þ so that The total pressure drop is given by the sum of the static pres- So S S u sure drop (i.e. due to the height difference), the momentum pres- q þð e oÞ ðq qoÞ : ¼ c ðE 12Þ sure drop (due to acceleration of the flow when it converts to gas q So S þðSe SoÞuðq q Þ o o or liquid) and the frictional pressure drop. < with a solution when q qo The static pressure drop is given by q So D < : : ps ¼ c 1 ðE 13Þ : ¼ qtpg sin h ðF 7Þ qo So S DL o > > o c > < o c As S 0 this occurs when So S þ S . When q qo or So S þ S where g is the gravitational acceleration and h is the angle of the container with respect to the horizontal. The two-phase density is q So þ S S ¼ e o > 1 ðE:14Þ given by [19] q S Sc o e : qtp ¼ qLð1 aÞþqV a ðF 8Þ c which has only a valid solution when Se > S . The above solutions < < where the void fraction is determined from the vapor quality, x exist when also qV q qL, otherwise after the step the content < > according to will be single phase vapor (q qV ) or single phase liquid (q qL). 1 x x 1 x UGA Appendix F. Pressure drop calculations a ¼ Co þ þ ðF:9Þ qV qV qL / F.1. Pure vapor flow with ! 0:25 2 2 For pure vapor flow the liquid flow is absent and the total flow gDqL A Co ¼ 1 þ 0:2ð1 xÞ is considered single phase flow. In such a case the friction coeffi- /2 cient for a pipe is given by the Darcy friction factor. This is only dependent on Reynolds number and relative roughness of the sur- and face. If Re < 2100: : g 0 25 : r qV 64 UG ¼ 1 18 1 f ¼ ðF:1Þ qL qL Re where r is the surface tension of the liquid. This holds for vertical otherwise: ()"#up flow. For vertical down flow the sign of UG is changed. 2 2:18 Re The momentum pressure difference per element length f ¼ 2 log þ ln ðF:2Þ between input and output is given by [19] 3:71D Re 1:816 ln 1:1Re lnð1þ1:1ReÞ 2 2 D = 2 where is the surface roughness, D the effective diameter and Re is Dp / ðA qtpÞ / mom ¼ ¼ f ðF:10Þ Reynolds number given by DL DL mom 2 2DA qV q v D / D Re ¼ V V ¼ V ðF:3Þ where gV gV A no 2D = = where q is the vapor density, v is the vapor velocity, g is the f mom ¼ qV qtp qV qtp V V V DL out in vapor dynamic viscosity and A the effective area of the pipe cross 42 V.-O. de Haan et al. / Cryogenics 85 (2017) 30–43
< 2D 6 6 where W = 500 kg/m2s and B z ; z is given by so that jf momj DL as qV qtp qL. So, the maximum momentum o ð 1 2Þ =D pressure drop corresponds to a friction factor of 2D L where D is 0 < z < 9:5 z > 3:8 p4ffiffiffiffiffiffiffiffi:8 2 1 : the diameter of the element and DL its length when the vapor qual- 3 8z1 4:8 ity changes from 0 to 1. A high heat input is needed to change the 1 < z1 6 3:8 z1 vapor quality so drastically. The heat input is mainly applied to z1 6 14:8 the moderator cell, while in the rest of the loop the heat input is rel- atively small. Hence, the momentum pressure drop (and the influ- 23:ffiffiffiffi4 9:5 < z2 < 28 z1 6 1:2 p ; : ence of the heat input on the friction pressure drop in the z2 z1 ðF 23Þ : elements) can be ignored. These are only relevant for the moderator > : 23pffiffiffiffiffi4 z1 1 2 : cell, where the ratio of 2D=DL is of the order of 1, comparable to the z2 1 2 friction factor of the inlet or outlet of the cell. The flow resistance 598 z > 28 z 6 1:2 pffiffiffiffi 2 1 z2 z (be it frictional or momentum transfer) of the moderator cell is very 2 1 hard to predict analytically and here a practical model has been z > 1:2 598pffiffiffiffiffi 1 z2 1:2 assumed as described in Appendix A. 2 The frictional pressure drop can be calculated using the Friedel and for the Lockhart-Martinelli correlation correlation model [19] applied to two-phase flow through a pipe. The frictional pressure difference per element length between ele- U 2 2 f ðð1 xÞReLÞ : LM ¼ð1 þ CXtt þ XttÞð1 xÞ ðF 24Þ ment and input/output is given by f ðReLÞ
2 2 Dp q v / : : : ¼ f V V ¼ U f ðF:11Þ where X ¼ ðÞx=ð1 xÞ 0 9ðÞq =q 0 5ðÞg =g 0 1; f ðReÞ is the friction DL 2D fr L 2 tt L V L V 2DA qL factor corresponding to Reynolds number Re and C is given by where the Darcy friction factor f is determined as before by using L ð1 xÞReL > 2300 xReV > 2300 C ¼ 20 the Reynolds number xRe 6 2300 C ¼ 10 V : : D/ ðF 25Þ ð1 xÞReL 6 2300 xReV > 2300 C ¼ 12 ReL ¼ ðF:12Þ gLA xReV 6 2300 C ¼ 5 with the two-phase multiplier given by In case of two-phase hydrogen flow of 3 g/s at 23 K, Müller- : U 3 24FH : Steinhagen and Heck correlation gives the smallest two-phase fr ¼ E þ : : ðF 13Þ Fr0 045We0 035 multiplier, then Friedel, Chisholm and Lockhart-Martinelli. where For the two-phase heat transfer coefficient we use q f : E ¼ð1 xÞ2 þ x2 L V ðF:14Þ htp ¼ hLð1 aÞþhV a ðF 26Þ qV f L
kLNuL kV NuV : 0:224 h ¼ and h ¼ ðF:27Þ F ¼ x0 78 þð1 xÞ ðF:15Þ L D V D
: : : 0 91 0 19 0 7 where NuL and NuV are determined by using the Dittus-Boelter rela- qL gV gV : H ¼ 1 ðF 16Þ tion with ReL; PrL and ReV ; PrV respectively. qV gL gL 2 1 / : References Fr ¼ 2 ðF 17Þ gDqH A [1] European Spallation Source; 2016.
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