radiation damage 318 doi: /s J. Synchrotron Rad. (2005). 12, Introduction

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1 Journal of Syncrotron Radiation ISSN Received 3 Marc 2004 Accepted 5 January 2005 Tree-dimensional numerical analysis of convection and conduction cooling of sperical biocrystals wit localized eating from syncrotron X-ray beams Asutos Maisekar, Micael J. Kazmierczak* and Rupak Banerjee Department of Mecanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, OH , USA. mike.kazmierczak@uc.edu Te differential momentum and termal energy equations for fluid flow and convective eat-transfer around te sample biocrystal, wit coupled internal eat conduction, are solved using advanced computational fluid dynamics tecniques. Average as well as local values of te convective eat-transfer coefficients are obtained from te fundamental equations. Te results of tese numerical solutions sow te tree-dimensional fluid flow field around te sample in conjunction wit te detailed internal temperature distribution inside te crystal. Te external temperature rise and maximum internal temperature increase are reported for various cases. Te effect of te important system parameters, suc as gas velocity and properties, crystal size and termal conductivity and incident beam conditions (intensity and beam size), are all illustrated wit comparative examples. For te reference case, an external temperature rise of 7 K and internal temperature increase of 0.5 K are calculated for a 200 mm-diameter cryocooled sperical biocrystal subjected to a 13 kev X-ray beam of potons s 1 mm 2 flux density striking alf te sample. For all te cases investigated, numerical analysis sows tat te controlling termal resistance is te rate of convective eat-transfer and not internal conduction. Termal diffusion results in efficient termal spreading of te deposited energy and tis results in almost uniform internal crystal temperatures (T internal 0.5 K), in spite of te non-uniform wit no more tan 1.3 K internal temperature difference for te worst case of localized and focused beam eating. Rater, te major temperature variation occurs between te outer surface of te crystal/loop system and te gas stream, T s T gas, wic itself is only about T external 5 10 K, and depends on te termal loading imposed by te X-ray beam, te rate of convection and te size of te loop/ crystal system. # 2005 International Union of Crystallograpy Printed in Great Britain all rigts reserved Keywords: beam eating; termal modeling; temperature increase; eat-transfer. 1. Introduction Te problem of eat transfer from X-ray eated biocrystals as attracted crystallograpers attention in recent years. Subjecting te biocrystal to a tird-generation syncrotron X-ray beam results in bot termal loading and radiation damage to te crystals. Cryogenic cooling of te biosample as been sown to elp alleviate te radiation damage problem to a great extent and terefore as become standard practice (Hope, 1990; Rodgers, 1994; Garman & Scneider, 1997; Garman, 1999). Unfortunately, it as been sown (as reported in past radiation damage worksops and in te recent literature) tat specific molecular structural canges still occur to te macromolecules wen exposed to tird-generation sources, even wen eld at cryogenic temperatures (Weik et al., 2000). Hence, radiation damage is a very important area of ongoing researc tat involves many issues. Various different aspects of tis complex problem are dealt wit in great detail elsewere, in oter articles in tis issue. Te focus of tis study is on te convection and conduction cooling of a cryocooled biocrystal sample from a pure termal eat-transfer point of view. More specifically, te aim of tis present analysis is to accurately determine te external and internal maximum temperature increase, and te eat-transfer rate from te biocrystal to te cooling cryostream. Available termal models, for predicting temperature rise owing to te absorption of X-ray beam energy, range in sopistication from simple to more advanced metodology. 318 doi: /s J. Syncrotron Rad. (2005). 12,

2 Te very basic adiabatic analysis (Helliwell, 1992) is often used in predicting te maximum rate of temperature increase (K s 1 ) of a crystal of a given mass aving arbitrary sape. However, it does not consider te energy transport away from te biocrystal to te gas stream (i.e. te convection term is neglected). Suc a model is reasonable only in te initial stage of te X-ray beam exposure and cannot be used to determine te actual final temperature of te crystal. Kuzay et al. (2001) and Kazmierczak (2001) included te convective eat-transfer coefficient in te model for te first time, altoug an estimated value, to predict te temperature of te crystal at steady state along wit a more realistic temperature rise and rate of increase troug te entire eating process. Teir work considered te lumped and distributed termal models for an infinite plane layer, cube and rectangular flat plate considering two different orientations. Simultaneously, Nicolson et al. (2001) performed a tree-dimensional finiteelement analysis on a macromolecular crystal subjected to a tird-generation syncrotron X-ray source and obtained: (i) te internal steady-state temperature distribution; (ii) te outside temperature drop; and (iii) te transient temperature response immediately after te beam is turned on. Te sape of te macromolecular crystal and te surrounding moter liquor was approximated as an ellipsoid. Two different gases, N 2 and He, wit estimated eat-transfer coefficient values of 300 and 800 W m 2 K 1, respectively, were used in teir analysis. Te next advance in modeling eat transfer from macromolecular crystals was carried out by Rosenbaum & Kazmierczak (2002) and Kriminski et al. (2003). Tese studies featured a more precise analysis of te convective eattransfer coefficient from te biocrystal surface to te cooling gas stream based on te pysical (velocity) and termopysical (viscosity, density etc.) properties of te gas. Rosenbaum & Kazmierczak (2002) approximated te biocrystal/ moter-liquor geometry as a flat disc and obtained a onedimensional analytical steady-state solution for te temperature distribution in te system as a function of te radius of te disc in te area illuminated by te beam, and in te region beyond. Te convective eat-transfer coefficient tat tey used was calculated from Witaker s (1972) correlation for a spere. Tis correlation uses gas velocity and gas fluid properties as parameters and is based on extensive experimental data. Kriminski et al. (2003) teoretically determined te convective eat-transfer coefficient (and its dependence on various gas flow parameters) by applying te boundary layer teory for viscous flow. Tey approximated te crystal surface as a flat plate to determine te external temperature rise. To obtain te steady-state internal temperature distribution, te one-dimensional eat conduction equation for speres was employed. It sould be noted tat all te termal models cited tus far assumed te convective eat-transfer coefficient to be constant over te entire surface of te biocrystal (ellipsoid, disc and spere), wic is not te case in reality owing to te complex gas flow field around te crystal loop geometry. Te present analysis goes beyond previous work to accurately obtain [via computational fluid dynamics (CFD)] te spatial Table 1 Te values of te termopysical properties (density, specific eat capacity, termal conductivity and viscosity) of N 2 and He gases at 100 K and 30 K, respectively, used in te numerical computations. Gas properties N 2 at 100 K He at 30 K (kg m 3 ) c p (J kg 1 K 1 ) k (W m 1 K 1 ) (kg m 1 s 1 ) Figure 1 Scematic of te system of interest. Te biocrystal, treated as a spere, is exposed to an X-ray beam wit absorption causing internal eating q 000 abs. Te source beam diameter is sown smaller tan te sample size (D b < D s ) but may be larger and irradiate te entire spere (D b = D s ). Te spere is immersed in a cooling gas stream wit uniform upstream velocity U 1 and constant temperature T 1. variation of over te surface of te spere, and tus allowing for te outer surface temperature to cange accordingly wile simultaneously calculating te temperature distribution witin te biocrystal. A complete parametric study is also performed by varying te pysical properties of te cryostream (velocity and gas type), and beam parameters (intensity and size), for various crystal sizes, to obtain te corresponding eat-transfer rate and te maximum internal and external temperature drops. 2. Matematical formulation Te biocrystal and moter liquor geometry, approximated as a spere, is subjected to an incoming X-ray beam and is convectively cooled in a stream of cold gas as sown in Fig. 1. Given tis configuration, te spere is internally eated owing to energy deposition and is externally cooled by convection wit te cold gas stream. Two different gases, N 2 or He at different temperatures, 100 K and 30 K, respectively, are used to cool te sample. Te termopysical properties of N 2 and He gases are given in Table 1 at teir respective temperatures. Te target diameter of te incoming X-ray beam, D b, can be reduced (focused beams) to eiter 10, 25 or 50% of te projected diameter of te spere, D s, or can be made te same size as te projected diameter of te spere (full beam). Te cold gas stream is treated as incompressible viscous flow and te upstream flow is assumed steady and unidirectional (i.e. constant uniform inlet velocity profile wit single velocity component). Te cooling gas stream (from jet nozzle outlet) is at a constant temperature T 1 and a velocity of U 1. Te treedimensional domain and finite volume mes, as sown in Fig. 2, was selected for te flow and conjugate eat-transfer J. Syncrotron Rad. (2005). 12, Asutos Maisekar et al. Sperical biocrystals 319

3 Figure 2 Computational domain used in te CFD analysis sowing te finite volume mes. Te flow domain extends 20 spere diameters downstream from te solid spere in te axial flow direction and ten diameters in te direction normal to te flow direction (xy plane). Finer grid spacing is used in and near te spere surface and in te wake region for greater accuracy. analysis witin and around te spere. Toug muc researc for flow over speres in te past as dealt wit a two-dimensional axisymmetric domain wic uses te vorticity and stream function formulation approac, tis study is based on te primitive variables formulation using velocity, pressure and temperature as primary degrees of freedom. Te continuity and momentum equations for te flow field around te spere are as follows, rðvþ ¼0; ð1þ rðvvþ ¼ rp þrðþþ g; were is te density, v is te velocity vector, p is te static pressure, is te stress tensor and g is te gravitational body force. Te stress tensor is described as ¼ rvþrv T ; ð3þ were is te molecular viscosity. Te differential termal energy equation tat gives te temperature field around te spere is of te form c p ðv rtþ ¼ k f r 2 T; ð4þ were k f is te termal conductivity of te fluid, c p is te eat capacity and is te density of te fluid. Te eat conduction equation in te solid region (spere) is given by k s r 2 T S ðx; y; zþ ¼0; were k s is te termal conductivity of te solid, T is te temperature and S is te volumetric internal eat source (discussed separately in te following subsection). Specific boundary conditions are required to complete te formulation. ð2þ ð5þ Flow boundary conditions are as follows: (i) no slip boundary condition is considered on te wall of te spere; (ii) U x = U 1, U y = 0 and U z = 0 at te inlet of te domain (uniform flow); (iii) U x = U 1, U y = 0 and U z = 0 on all te lateral surfaces of te external flow domain; (iv) stress free at te outflow of te domain wit gauge pressure being zero. Termal boundary conditions: (i) T = T 1, constant temperature at inlet of te fluid flow domain; (ii) T = T 1 on te lateral surface of te flow domain; (iii) at outlet te temperature gradients in te direction of te flow are set to zero; (iv) continuity of temperature and eat flux on te surface of te spere. Te finite volume mes was developed wit exaedral elements. Te mes was graded wit a finer spacing in and around te wake region of te spere so as to accurately model te flow and temperature fields. Te solid spere was represented by a total number of exaedral elements wereas exaedral elements were used in te flow domain surrounding te spere. Te domain size was cosen suc tat te lengt was 25 times and te widt and eigt were 10 times te diameter of te spere. Typical computational run time on a Pentium 4, 2.4 GHz wit 1024 MB RAM, was about 3. More information regarding code validation and oter numerical details can be found by Maisekar et al. (2005) Heat source distribution Te internal eat source distribution witin te spere (Fig. 3) depends on te local absorption of te source beam, wic, in turn, depends on te intensity of te source beam, dept of target and material caracteristics (crystal composition). Matematically te source term S depends strongly on spatial location and is given by Figure 3 Internal eat source distribution inside te spere wen exposed to an X-ray beam of flux potons s 1 irradiating 50% of te spere. (a) q 000 abs contours (in W m 3 ) plotted on te axial yz plane passing troug te spere center for L att = 3.9 mm. (b) Axial profiles for all tree different values of L att studied. 320 Asutos Maisekar et al. Sperical biocrystals J. Syncrotron Rad. (2005). 12,

4 Table 2 Summary of runs. Case 1 serves as te reference case wit results igligted in italic in all subsequent tables. Parameters are varied as sown in runs 2 16 to illustrate teir effect on te resulting eat transfer and crystal temperatures. Case number Gas velocity (m s 1 ) k (W m 1 K 1 ) Intensity (potons s 1 mm 2 ) S ðx; y; zþ ¼q 000 gen ¼ ðio=l 00 att Þexpð L=L att Þ n ¼ ðio=l 00 att Þexp z ðd s =2Þ 2 x 2 þ y 2 L att (mm) 1=2=Latt Beam diameter (mm) were Io 00 is te incident intensity of te source beam, L is te dept of target (distance traveled troug te spere), L att is te beam s attenuation lengt (material property tat depends on te energy of te beam) and D s is te diameter of te spere. Fig. 3 sows te variation in energy deposition along a plane passing troug te center of te biocrystal for te base case. Te beam source used ere was a focused beam wit a flux density of potons s 1 mm 2 and L att = 3.9 mm (reference case) striking alf of te biocrystal, i.e. D b = D s /2, and terefore potons s 1 total flux into a 0.2 mm-diameter spere. Te source profiles for two oter values of L att, representing ig and low values, are also plotted (to be considered later to determine its impact on temperature distribution). Besides absorption lengt, anoter important consideration in te source term is te size (i.e. target area) of te source beam. For te full beam, te source beam diameter is equal to te diameter of te spere (0.2 mm for te base case) wereas for te focused beams te diameter of te source beam is reduced to alf, one-quarter and onetent of te diameter of te spere, and tus te flux densities are four, 16 and 100 times greater, respectively. However, it sould be noted tat te total incident power striking te spere (flux into sample) is kept constant (I0 00 A B = potons s 1 ) for all cases. Tis is acieved by increasing te incident intensity in te focused beam according to te following relation, I0 00 FullA Full I0FcA 00 Fc ¼ 3: potons s 1 ; ð7þ were A Fc, A Full and I0Fc, 00 I0 00 Full are te areas and incident intensities of te focused and full beams, respectively. For te o ; ð6þ Crystal diameter (mm) N N N N N N N N N N N N N N N He focused beam, te power is absorbed in a cylindrical region passing troug te spere center. 3. Results and discussion Te sample consisting of crystal and moter liquor, approximated as a spere, subjected to a tird-generation X-ray beam is analyzed using te abovementioned tree-dimensional numerical finite volume model. Te salient objectives of te present analysis are to accurately obtain te flow field around te biosample for a given gas velocity, surrounding temperature field, internal temperature distribution witin te biosample and rate of convective eat transfer from te biocrystal to te cold stream. In all, 16 different cases were studied (see Table 2). Te internal and external values of T for various cases are compared and te cange in convective eat-transfer coefficient is studied by varying te gas velocity (runs 1 4), termal conductivity of te crystal (runs 5 6), canging te intensity of te source beam (runs 7 8), altering te absorption lengt (runs 9 10), focusing te beam to smaller and larger size (runs 11 13), increasing te crystal size (for a constant beam size, runs 14 15) and, finally, canging te gas type (run 16). Te results for te baseline case (run #1, in Table 2) give te complete details of te flow and eat transfer in and around te biocrystal for te given set of parameters. Tey are discussed first in dept and serve as te reference case for all te oter runs investigated. 4. Baseline case results Te sperical biosample in te loop size of D s = 200 mm is convectively cooled by te N 2 gas stream flowing over te sample at 100 K wit a velocity of U gas =1ms 1. Te source beam is a focused X-ray beam of size D b = 0.1 mm (50% of D spere ), wit an intensity of potons s 1 mm 2 at 13 kev. For L att = 3.9 mm te amount of energy absorbed in te biocrystal is equal to q abs = mw. Te termal conductivity of te biocrystal, k spere, is taken to be 0.6 W m 1 K 1. Te parameters of te termopysical properties of N 2 gas at 100 K, used in te computations, are sown in Table 1. Te flow field around te biocrystal, as depicted by te velocity vectors at mid-dept as viewed from te normal to te flow direction (YZ plane) at steady state, is sown in Fig. 4(a). Te velocity vectors are color-coded wit maximum velocity sown in red and minimum velocity in blue. Tere exists a complex flow field around te spere and tere are large variations in velocity in te immediate vicinity of te spere s surface. A recirculation region is evident beind te spere wit te formation of a single axisymmetric donut-saped J. Syncrotron Rad. (2005). 12, Asutos Maisekar et al. Sperical biocrystals 321 Gas type

5 vortex owing to te combination of viscous sear forces and adverse pressure gradient caused by te sperical sape. Te lengt of tis region is dependent on te upstream flow velocity, or non-dimensional Reynolds number defined as Re = VL/ were, V and are te density, velocity and viscosity of te fluid, respectively, and L is te caracteristic lengt of te body, being te diameter D spere in tis case. Fig. 4(b) sows te temperature variation in te flowing gas stream surrounding te sperical biocrystal. Te energy tat is absorbed by te biosample owing to exposure to te X-ray beam must first be conducted to te outer wall, and is ten carried away by te gas stream, as indicated by te temperature variation in te gas stream. Te temperature gradients near te spere surface are very large, especially in te very slender region near te front alf of te spere tat forms te so-called termal boundary layer. Te temperature at te outer wall of te spere is also sown and te rise in average wall temperature, T wall, above te free stream N 2 gas temperature owing to energy absorption is about 7 K. Te local eat-transfer coefficient,, varies spatially over te surface of te spere because of te complex flow pattern. Fig. 5 sows plotted against te angular displacement along te surface of te biocrystal. Te local eat transfer varies from a maximum value of 614 W m 2 K 1 at te stagnation point at te front of te spere to 130 W m 2 K 1 at te point of flow separation ( 140 ) before increasing sligtly again at te rear of te crystal. Tis is due to te flow field near te spere surface, wic results in a maximum of te normal velocity gradient near te stagnation point. Gradually te velocity and its gradient reduce to zero at te point of flow separation. Away from te flow separation point te velocity increases again owing to te flow recirculation in te wake. Te average convection eat-transfer coefficient is calculated by integrating te local value over te entire surface of te sperical crystal and is found to be = 346 W m 2 K 1. Te temperature contours inside te sample at mid-dept from te side (YZ plane) and front (XY plane) are sown in Figs. 6(a) and 6(b), respectively. Te energy is almost uniformly absorbed inside te central cylindrical core region of te biocrystal owing to te relatively large value of te absorption lengt of te source beam. Te final steady temperature distribution sown is te result of te energy balance between te diffusion of te deposited energy (eat conduction) inside te solid and tat convected from te outer surface. Fig. 6(a) indicates iger temperature in te rear of te biocrystal wic is due to te lower local convective eattransfer rate,, as sown in Fig. 5. Te maximum internal Figure 5 Variation of local along te surface of te spere as calculated for te reference case. Te temperature field results in a maximum value at te front stagnation point, = 0, and a minimum value occurs at te point of flow separation, 140. Figure 4 Numerical computations for te reference case: (a) te flow field past te spere depicted by velocity vectors; (b) te temperature field in te gas stream surrounding te spere as illustrated by isoterms. = 7.2 K. Te velocity field sows te flow separation and te large recirculation region downstream of te spere. Note tat te termal field and temperature gradients immediately before and after te spere are considerably different owing to te presence of tis large wake. Figure 6 Temperature contours (isoterms) inside te spere for te reference case plotted at mid-dept: (a) side view (YZ plane) and (b) front view (XY plane). = 0.56 K. Te iger temperature in te rear of te spere is due to lower local in te wake region. However, note tat te temperature field inside te spere plotted on te XY plane is axisymmetric (circumferentially symmetric in te direction normal to te flow stream altoug asymmetrical in te longitudinal flow direction). 322 Asutos Maisekar et al. Sperical biocrystals J. Syncrotron Rad. (2005). 12,

6 Table 3 Te effect of gas velocity on, T outside and T internal wit all oter parameters eld at te reference case values. Wit increasing velocity, increases and terefore reduces T outside. However, T internal remains practically constant and independent of gas velocity. Velocity (m s 1 ) (W m 2 K 1 ) temperature difference in te biosample, T internal = T max T wall, in tis case is only 0.56 K and is muc less tan te average external temperature rise in te biocrystal wic was given as T outside = T wall T gas = 7.16 K in Fig. 4(b). 5. Effect of te cooling stream velocity Figure 7 Te local eat-transfer coefficient versus calculated for various gas stream velocities. Comparison sows tat te local value of increases as gas velocity increases at all locations except at te flow separation point. Increasing te gas stream velocity improves te rate of convective eat transfer from te biocrystal surface to te gas stream. Fig. 7 sows te local variation of over te surface of te spere for tree different values of gas velocity. As te gas velocity increases, te local and average convective eattransfer coefficients increase. Te value at te stagnation point is greater for iger velocities and reduces over te surface of te crystal until te point of flow separation, and is about te same at tat point for all tree different gas velocities. Te increase in local in te back region of te crystal is dependent on te strengt of te recirculation velocity, wic in turn is dependent on te free-stream velocity. In steady laminar flow, te iger te upstream velocity te larger te recirculation zone and te stronger te recirculation velocities and gradients, and tus te local value of is iger in tat region. A review of te relevant fluid mecanics literature sows (Lee, 2000) tat te recirculation region remains attaced and symmetric about te axis passing troug te center of te spere up to a maximum Reynolds number of Re = 220 (U m s 1 for 0.2 mm spere Table 4 Effect of termal conductivity k on, T outside and T internal. Te cange in k affects only T internal, wic decreases wit increasing k. A cange in k does not appreciably alter eiter or T outside. k (W m 1 K 1 ) cooled by N 2 gas), and remains attaced but asymmetric for 220 < Re 350, wile still witin te laminar flow regime. For Re > 350, te flow starts sedding wit oscillating alternating vortices tat eventually becomes unstable and transition to turbulence occurs. Table 3 sows te variation in te average eat-transfer coefficient,, te external temperature rise, T outside =, and te internal temperature difference, T internal =, for four different velocities. increases wit increasing velocity (second column) and, as a result, te external temperature rise T outside (tird column) decreases. Te variation in flow velocity does not alter te maximum internal temperature difference and T internal is almost te same for all of te stated velocities (last column). Rater, T internal depends on te rate of internal eat conduction and, in particular, on te value of te termal conductivity of te biocrystal, as will be sown in te following section. 6. Effect of k spere (W m 2 K 1 ) Table 4 sows te effect of varying te termal conductivity, k spere, of te biocrystal and sows tat te cange in te termal conductivity of te biocrystal affects only te internal region of te biocrystal, i.e. T internal, wereas, attributed to convection, essentially remains constant. As k spere increases, T internal decreases rougly by te same order of magnitude (last column). Te external temperature rise, T outside, remains almost te same for te tree different k values since all calculations produce a similar, as a result of uncanged flow caracteristics (same U 1 ) and fixed eatsource parameters. 7. Effect of varying beam parameters Anoter important objective is to analyze te eat transfer under varying beam conditions, specifically different intensity, attenuation lengt and beam size. Variable beam intensity, I0 00, is taken into consideration in te present analysis in Table 5. Detailed calculations sow tat tere is no effect on te average eat-transfer coefficient (fourt column) wit te cange of beam intensity. However, te outside temperature difference (second column) and te maximum internal temperature difference (tird column) cange significantly. Te increase in beam intensity raises T outside and T internal by rougly te same order of magnitude (i.e. temperature J. Syncrotron Rad. (2005). 12, Asutos Maisekar et al. Sperical biocrystals 323

7 Table 5 Varying beam intensity. Te intensity of te beam is canged, keeping te beam area constant and its effect on T outside, T internal, and q deposited is sown. Wit increasing intensity, T outside, T internal and q deposited increase, but tere is no cange in. I 00 0 (potons mm 2 s 1 ) (W m 2 K 1 ) q (W) increase is rougly linear wit beam intensity). Table 6 sows ow te variation in L att affects temperature. Te variation in L att is due to eiter different incident beam energy or canges in material properties. As L att decreases, q absorbed increases (last column) and terefore tere is a corresponding increase in bot te external temperature rise, T outside, and te internal temperature difference, T internal. Again remains approximately te same owing to te uncanged fluid flow properties and almost isotermal surface wall temperature. Te effect of beam size (Table 7) is investigated relative to te reference case (50%) by eiter expanding it to full beam diameter (D b = D s or 100%) or by focusing it down to 25% or 10% diameter, wile keeping te incident power constant. Tis was acieved by decreasing (or increasing) te incident intensity for te full (or focused) beam as discussed earlier. A relatively small cange in te maximum internal temperature difference is observed (column 2) wit te cange in beam size; te internal temperature difference, T internal, reduced from 1.3 K for te 10% beam to 0.2 K for te 100% (full) beam but overall te magnitude of te internal temperature difference is still rater small relative to te outside temperature increase. Te external temperature rise, T outside, (tird column) reduced from 7.6 K to 5.0 K by canging from te 10% to te 100% beams, respectively. It can be observed tat te external temperature difference is almost constant for all of te tree focused beams (i.e. 10%, 25% and 50% beams) owing to te fact tat te power absorbed is almost identical for tese tree cases (last column), and since (fourt column) remains te same. However, in te case of te 100% (full) beam, tere is less power absorbed in comparison wit te focused beams, even toug te incident power, i.e. A B I0, 00 is kept fixed in all four calculations. Tis is due to te large variation in te absorption pat lengt for te source beam over te surface of te sperical biocrystal, i.e. absorption dept reduces to zero at bot te top and bottom. Less total energy is absorbed and te similar results in a smaller external temperature rise. Fig. 8 sows plots of te internal temperature distribution inside te biocrystal at te mid-dept from te side (Fig. 8a) and te front (Fig. 8b), canging from focused (10%) to full (100%) incident source beam sizes. Te side-view contours for 10% source beam size clearly sow iger temperatures in te cylindrical region in wic te energy from te source beam is absorbed compared wit te rest of te spere. Also, it can be noted tat te otter region is sifted towards te rear of te Table 6 Effect of varying absorption lengt, L att. Te cange in L att affects T outside, T internal and q deposited, wit remaining practically constant. T outside, T internal and q deposited decrease wit increase in absorption lengt. L att (mm) (W m 2 K 1 ) q (W) Table 7 Effect of te size of te X-ray beam. A large cange in beam size (maintaining same total flux) results in only modest cange in T internal and does not affect. Te major temperature difference is still T outside wic depends on and on te total amount of q absorbed. Beam size (W m 2 K 1 ) q (W) 10% % % % biocrystal owing to lower local convective eat-transfer coefficient,, in te wake. For te full-beam case, te isoterms (lines of constant temperature) appear more circular in sape, owing to te fact tat energy is almost uniformly deposited over te entire sperical region in te biocrystal. As te beam is focused te temperature contours are more localized near te central core region. However, irrespective of te extent of localization, te energy absorbed is redistributed over te entire sperical region owing to termal diffusion. Te maximum temperature attained in eac case decreases as te beam area increases, owing to almost te same amount of energy being distributed over a larger cylindrical beam region and because of closer proximity to te convectively cooled exterior surface. Fig. 8(b) sows te temperature distribution inside te biocrystal at te mid-dept from te front. Te isoterms form concentric rings wit closer spacing of contours concentrated (i) in te central cylindrical region in wic te focused beam energy is absorbed, and also (ii) at te surface in te surrounding termal boundary layer. Te sarper temperature gradients located outside te surface are due to convection eat transfer. Figs. 9(a) and 9(b) sow te axial temperature profiles along te centerline of te biocrystal in te Z (side view, from front to back) and X (front view, from left to rigt) directions, respectively. From te side profiles (top plot) it can be clearly seen tat te maximum centerline temperature, T max, is sifted towards te rear of te biocrystal. Tis is attributed to te fact tat te local eat-transfer coefficient,, is lower in tat region. Also, te magnitude of T centerline is greatest for 10% beam (solid line) and reduces as te beam size increases to 25% and 50% (dased lines), owing to similar amounts of energy being deposited into a larger region tat is closer to te gas stream aving lower temperature. Also sown is T centerline 324 Asutos Maisekar et al. Sperical biocrystals J. Syncrotron Rad. (2005). 12,

8 Table 8 Effect of crystal size, keeping all oter parameters fixed. An increase in crystal size increases te q deposited. However, owing to an increase in and a large increase in surface area of te crystal, T outside decreases. Te cange in T internal is almost negligible. D spere (mm) Area (m 2 ) (W m 2 K 1 ) q (W) for te 100% beam (dot-dased line) case, wic is considerably lower tan te focused-beam results, owing to te reduced amount of absorbed energy caused by te overall sorter absorption pat lengt at te top and at te bottom of te sperical biocrystal. Fig. 9(b) sows T centerline profiles in te x direction. Tis plot again sows tat te maximum temperature is igest for te 10% beam but diminises (and te profile spreads out ) wit te increase in beam size. T max peaks at te exact geometric center of te plot because of te symmetry of te fluid flow field and te eat source distribution in te x direction, across te flow stream. Finally, superimposed in Figs. 9(a) and 9(b) are te temperature profiles calculated for te case of a sperical biocrystal obtained from a simpler one-dimensional termal model assuming a uniform convective eat-transfer coefficient (i.e. spatial variation neglected). Te amount of energy deposited, q, used in tis simpler analytical model is set equal to te same amount tat is absorbed in te 100% beam size case aving non-uniform, but is evenly distributed trougout te spere. Also, te convective eat-transfer coefficient assumed ere is set equal to te average convection eat-transfer coefficient calculated over te entire surface of te biocrystal from our CFD model. Tis average value is te same value everywere, tus rendering tis termal model truly one-dimensional and permitting a very easy analytical solution (Kriminski et al., 2003). It can be seen tat te temperature profiles generated from tis simplified model (bottom solid line in bot upper and lower plots) are symmetric in bot side and front views (z and x directions, respectively). Also, te values of T centerline for te onedimensional case are very similar in magnitude to te more advanced numerical solution for te 100% beam case wit non-uniform calculated over te surface of te biocrystal. Tus, te difference between te two sets of lines can be attributed mainly to te variation in local and is not very large if te energy is deposited trougout te spere (i.e. full beam). However, it is expected tat te differences between te results generated by te two different models will become more pronounced as te X-ray beam is increasingly focused. Table 9 Comparison sowing N 2 versus He gas cooling (for same gas jet velocity). He at 30 K results in tree times larger tan N 2 at 100 K and terefore a lower T outside. Te iger termal conductivity of te crystal at lower temperature is responsible for te smaller T internal sown. Gas (W m 2 K 1 ) N 100 K K Effect of crystal size Table 8 and Fig. 10 sow te effect of increasing te crystal diameter from 0.2 mm to 0.4 mm to 0.8 mm, keeping te source beam target area, beam intensity and velocity of te N 2 gas stream te same in all tree cases. Numerical computations sow tat te average eat-transfer coefficient,, increases (column 3) from 346 to 458 W m 2 K 1 wit increasing crystal size. Moreover, tere is a very large increase (16) in te surface area (column 2) of te biocrystal. Bot of tese factors will enance te rate of convective eat transfer and lower te crystal temperature. However, coupled wit tis, te energy deposited in te biocrystal increases wit increasing crystal size (last column) owing to te longer absorption pat lengt, wic will raise te temperature of te sample. Wit all of tese factors taken into account, te numerical calculations sow tat, as te crystal size increases, te outside temperature difference, T outside (fourt column), actually decreases from about 7.2 to 5.9 K, but te maximum internal temperature difference, T internal (fift column), remains almost constant. Altoug te amount of energy deposited rises wit increasing crystal size, te increase in average convective eat-transfer coefficient,, and greater surface area dominate, resulting in a lower outside temperature difference, T outside. Fig. 10 sows te internal temperature contours for all tree crystal diameters, 0.2, 0.4 and 0.8 mm, wen exposed to te same X-ray beam. Clearly te temperature is igest in te cylindrical region in wic te energy is deposited. Energy is ten conducted away troug te rest of te spere volume to its outer wall, but still te temperature differences inside te largest crystal are relatively small. Te rear of te spere is again otter compared wit te front owing to te relatively lower value of te local convective eat-transfer coefficient,, in te wake region of te biocrystal. A muc greater reduction in crystal temperature is possible if te system geometry can be canged suc tat te surface area is increased witout increasing te amount of energy deposited, for example by using larger and flatter (constant tickness) crystals. 9. Effect of gas properties Anoter numerical computation was performed by canging te gas coolant from N 2 at 100 K to He at 30 K as well as canging te termal conductivity of te biocrystal sample from 0.6 to 5 W m 1 K 1. It is expected tat te termal conductivity of te material increases wit decreasing temperature, ere from 100 K to 30 K (Dillard & Timmeraus, 1966; Klemens, 1969; Kaviany, 2002). Te various termopysical properties for bot N 2 and He gases at 100 K and 30 K, respectively, are listed in Table 1. Table 9 sows te difference in average eat-transfer coefficient, T outside and J. Syncrotron Rad. (2005). 12, Asutos Maisekar et al. Sperical biocrystals 325

9 T internal (second, tird and fourt columns, respectively) owing to canging te gas coolant from N 2 to He. Te numerical simulations reveal tat for He gas is approximately tree times iger tan tat of te N 2 gas (1078 versus 346 W m 2 K 1 ). Tis results in a proportionally sarp reduction in external temperature difference of T outside = 7.2 K for a biocrystal wit N 2 at 100 K to T outside = 2.3 K wen using He at 30 K. Also, te internal temperature difference, T internal, is muc lower compared wit te N 2 gas case, not owing to te enanced rate of convection but rater because of te iger termal conductivity of te material sample used in te conduction analysis. As presented earlier, T internal is inversely proportional to te termal conductivity of te material and ence an increase in termal conductivity of te material decreases T internal by approximately te same order. Te flow field surrounding te biocrystal calculated from te numerical analysis wen cooled wit He gas is sown in Fig. 11. It appears very similar in sape to te flow pattern described in Fig. 4(a) for N 2 gas cooling, except tat te size of te recirculation zone located beind te spere is muc sorter in lengt. Likewise, comparison of te local convection Figure 9 Temperature profiles calculated for te different beam sizes (along wit analytical one-dimensional solution assuming uniform ): (a) side view; (b) front view. Te spere temperature increases as te X-ray beam is focused. Along te flow stream direction (z-direction profiles) te temperatures are iger at te rear of te spere, and are igest at te center of te spere (and symmetrical) wen traversing normal to te flow stream direction (x-direction profiles). Figure 8 Isoterms (constant temperature contours in K) inside te biocrystal for four different sizes of X-ray beam wit constant flux, sowing te internal temperature distribution dependence on te size of te source beam (i.e. on te flux density). Smaller more focused X-ray beams result in sarper internal temperature gradients (energy is deposited in a smaller region) and sligtly iger maximum temperatures, altoug te average bulk temperature of te spere remains fairly constant: (a) side view (YZ plane) and (b) front view (XY plane). Figure 10 Internal temperature distribution for different-sized crystals exposed to te same (reference) beam. Te target area irradiated by te source beam is eld fixed but more energy is deposited for ticker crystals. However, te maximum internal temperature difference remains almost te same in all cases and te outside temperature difference decreases owing to te added surface area for convection. Refer to Table 8 for te actual temperature differences. 326 Asutos Maisekar et al. Sperical biocrystals J. Syncrotron Rad. (2005). 12,

10 Table 10 Summary of results from te parametric investigation. Te external and internal crystal temperature differences are reported over a range of conditions. Te parameters are listed in rank order, te total beam flux being te most important parameter and te beam size (flux density, assuming constant flux) te least, wit regard to teir impact on crystal temperatures. Te convection parameters and crystal size alter te external temperature difference (dominate temperature increase) wereas te internal temperature rise is always minor by comparison. Parameter T external T internal Figure 11 Velocity field for He gas stream flowing past te spere depicted by velocity vectors. Te flow pattern is similar to tat obtained using N 2 gas except tat te recirculation zone is sligtly sorter. Figure 12 Local variation along te surface of te spere for He gas cooling at 30 K plotted wit N 2 at 100 K for te same gas stream velocity, sowing significant increase in at all locations. Comparison sows tat He gas outperforms N 2 gas in terms of iger local and average eat-transfer coefficients. coefficients for te two different gases (Fig. 12) reveals very similar beavior in terms of spatial dependence, but sows te large difference in te magnitudes, essentially owing to te differing values of termal conductivity of te two gases (i.e. difference in gas properties and not flow patterns). q abs (flux, L att ) ( Linear) ( Linear) k spere No effect ( Linear) Gas type (N 2! He) (7! 2 K) (0.5! 0.05 K) Gas velocity (0.5! 2ms 1 ) (9! 6 K) Negligible Crystal size (0.2! 0.8 mm) (7! 6 K) Negligible Beam size (100! 10%) Negligible (0.2! 1.3 K) 10. Conclusions Te temperature increase during intense X-ray beam eating of sperical biocrystals as been carefully analyzed using advanced CFD modeling. Numerical solutions provided te following: (i) accurate local and values for convection; (ii) fluid flow and temperature fields surrounding te body; (iii) coupled internal temperature distributions witin te crystal. For a typical 0.2 mm-diameter biocrystal, subjected to an intense tird-generation 13 kev X-ray beam of potons s 1 focused on alf of te crystal, results sow tat T external = 7.16 K and T internal = 0.56 K. Te local eattransfer coefficient,, varied from 614 to 130 W m 2 K 1 over te surface of te spere and te average eat-transfer coefficient was = 346 W m 2 K 1. Using te numerical model, te investigation presented te effect of several parameters, suc as te gas stream velocity U 1, termal conductivity of te spere k spere, tree beam parameters (beam intensity I0, 00 absorption lengt L abs and beam size D b ), crystal size and te type of gas coolant, to obtain te expected temperature rise over a range of different operating conditions. Te comparison of results, in order of greatest to least importance, wit respect to bot external and internal temperature difference is sown in Table 10. Total termal load, convection rate and crystal size were te main controlling factors tat determined te sample temperature. Beam size ad less impact since internal eat conduction resulted in effective termal spreading of te deposited energy. It was sown tat, in general, te internal temperature rise witin small crystals is relatively small, i.e. T internal 0.5 K, and is about te same order of magnitude for bot full and focused beams owing to te efficient termal spreading by internal termal diffusion, i.e. eat conduction. Te major temperature increase is in te external temperature rise, T outside =, wic is about 7 K and is limited by te rate of convective eat transfer. It was sown tat using bigger sperical crystals (for fixed beam size) results in lower temperatures tan for smaller crystals owing to te added surface area for convection (but muc greater improvement is expected if te surface area for convection is increased witout increasing te absorption dept). Finally, a brief comparison of te more sopisticated tree-dimensional CFD results against te simpler one-dimensional model (uniform ) solution sowed tat te actual spatial variation in te convective eat-transfer coefficient (caused by te surrounding fluid flow field) results in sligtly elevated temperatures in te back region of te biocrystal. However, tis as only a rater minimal effect on te bulk crystal temperatures owing to te relatively small crystal size and efficient termal spreading by internal eat conduction. Hence it is concluded tat, in terms of simplified termal modeling of small crystals, one may reasonably calculate an approximate T outside using an average tat is obtained from an accurate J. Syncrotron Rad. (2005). 12, Asutos Maisekar et al. Sperical biocrystals 327

11 empirical convection correlation, and estimate maximum T internal using a simple one-dimensional eat conduction solution. 11. Recommendations for future work Te sape of te biocrystal surrounded by moter liquor was considered to be a spere, wic at best is only a roug approximation; more realistic geometry sould be modeled to accurately simulate fluid flow and convective eat transfer from actual crystal/loop systems. Te termopysical properties used in te present study are based on values from te prior literature, wic are estimates based on available resources and need to be more accurately determined. Te termal conductivity of te moter liquor and te crystal were taken to be te same; owever, differences between cryoprotectant mixtures and crystal properties sould be taken into account as well as peraps local impurities and possible non-omogeneities in te crystal itself. Transient temperature beavior is an important aspect of te problem tat needs to be studied, especially te time required to acieve steady-state conditions under continuous beam compared wit pulsedbeam operations. Te last, and peraps te most important, recommendation at tis time is to experimentally verify tese temperature predictions in a series of carefully controlled experiments at a participating syncrotron. References Dillard, D. S. & Timmeraus, K. D. (1966). Pure Appl. Cryogen. 4, Garman, E. (1999). Acta Cryst. D55, Garman, E. & Scneider, T. R. (1997). J. Appl. Cryst. 30, Helliwell, J. R. (1992). Macromolecular Crystallograpy wit Syncrotron Radiation, pp Cambridge University Press. Hope, H. (1990). Annu. Rev. Biopys. Biopys. Cem. 19, Kaviany, M. (2002). Principles of Heat Transfer, pp New York: Wiley. Kazmierczak, M. J. (2001). Second International Worksop on X-ray Damage to Crystalline Biological Samples, Argonne National Laboratory, Argonne, Illinois, USA. Abstract. Klemens, P. G. (1969). Termal Conductivity, Vol. 1, edited by R. P. Tye, pp London: Academic Press. Kriminski, S., Kazmierczak, M. J. & Torne, R. A. (2003). Acta Cryst. D59, Kuzay, T. M., Kazmierczak, M. J. & Hsie, B. J. (2001). Acta Cryst. D57, Lee, S. (2000). Comput. Fluids, 29, Maisekar, A., Kazmierczak, M. J. & Banerjee, R. (2005). Num. Heat Trans. J. Part A. In te press. Nicolson, J., Nave, C., Fayz, K., Fell, B. & Garman, E. (2001). Nucl. Instrum. Metods Pys. Res. A, 467/468, Rodgers, D. W. (1994). Structure, 2, Rosenbaum, G. & Kazmierczak, M. J. (2002). Acta Cryst. A58(Suppl.), C279. Weik, M., Ravelli, R. B. G., Kryger, G., McSweeney, S., Raves, M. L., Harel, M., Gros, P., Silman, I., Kroon, J. & Sussman, J. L. (2000). Proc. Natl. Acad. Sci. USA, 97, Witaker, S. (1972). AICE J. 18, Asutos Maisekar et al. Sperical biocrystals J. Syncrotron Rad. (2005). 12,