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electronic reprint Heat transfer from protein crystals: implications for ...

electronic reprint Heat transfer from protein crystals: implications for ...

esearch papersimportant

esearch papersimportant quantity averaged over the entire sample surfacegives a good description of the actual heat-transfer rates.In addition to the Reynolds number characterizing the ¯ow,heat transfer in ¯uids is characterized by the Prandtl numberPr = v/, where v is the ¯uid's kinematic viscosity and is itsthermal diffusivity. As shown in Table 1, for both N 2 and Hegas Pr is close to 1, independent of the gas temperature.From scaling analysis for ¯uids with Pr ' 1 and with Re L inthe laminar boundary-layer regime, the average heat-transfercoef®cient h over the surface of a spherical sample isapproximately given by (Landau & Lifshitz, 1987; Bejan,1995)h ' ˆ 0:6 u1=2 t …Lv† ;…3†1=2where t is the average thickness of the laminar thermalboundary layer given by t ' 1:7L=Re 1=2L ˆ 1:7…vL=u† 1=2 …4†and where is the thermal conductivity of the cooling ¯uid.Experiments and numerical simulations are in agreementwith (3) and (4) (Vyrubov, 1939; Liakhovski, 1940; Yuge, 1960;Whitaker, 1972; Bald, 1984; Wong et al., 1986). Heat-transferrates are larger (smaller) in the upstream (downstream) regionof the sample where the boundary layer is thinner (hasseparated) (Renksizbulut & Yuen, 1983; Sharpe & Morrison,1986; Wong et al., 1986; Bagchi et al., 2001; Balachandar & Ha,2001). Averaging over the sample surface yields the empiricalrelation (3). Although primarily derived for a sphericalsample, (3) is also roughly applicable to rod- or plate-likesamples with their long dimension oriented parallel to the ¯owand with L corresponding to this dimension. Based on ouranalysis, we also expect that h is nearly independent of thetemperature difference between sample and ¯ow.Table 3(b) gives estimates of h calculated using (3) forcrystals of different size cooled from room temperature bygases at different temperatures. These are in fair agreementwith values used in previous analyses: 100 < h < 250 (Kuzay etal., 2001) and 300 < h < 800 (Nicholson et al., 2001). Thesevenfold difference in thermal conductivity between He andN 2 gas is partially compensated by a threefold difference in thethickness of the boundary layers, so that h differs by only afactor of 3 for both gases.A similar approach yields estimates of h for plunge-coolingin cold liquids in the absence of boiling, also given in Table3(b). From Table 1, Pr ' 2 for liquid nitrogen, roughly threetimes that for nitrogen gas; Pr < 10 for ethane and propaneexcept extremely close to propane's melting point. The heattransfercoef®cient h is only weakly dependent on Pr, varyingas h ' Pr 0.3 ...0.4 (Kramers, 1946; Ranz & Marshall, 1952a,b;Whitaker, 1972; Bald, 1984). Thus (3) should be approximatelyapplicable for constant-velocity plunge-cooling in liquidcryogens, where u is now the plunge velocity.For He and N 2 gases, the temperature dependencies of thekinematic viscosity v and thermal conductivity cancel in (3)to make h essentially independent of temperature, so that thetemperature at which v and are evaluated does not matter.The situation is more complicated for liquid propane andethane, whose viscosity v increases steeply with decreasingtemperature. In this case, the boundary layer is roughlydetermined by the smallest kinematic viscosity, which occursin the warmest regions adjacent to the sample surface.Consequently, we use cooling ¯uid properties at the initialsample temperature T 300 K in our estimates of h for liquidsin Table 3(b), at suf®cient pressures so that they remain liquid.The critical temperature for nitrogen T = 126 K is well below300 K, so that nitrogen near the sample surface will initially bein an essentially gaseous state. To estimate the largest practicalcooling rates, we calculated h at T = 300 K at a pressureP = 10 MPa, three times nitrogen's critical pressure. Because hwill vary during cooling, these estimates will be less accuratethan those for gases. However, as discussed later it is usuallyimpractical to perform plunge-cooling with no boiling, so thatthese estimates are suf®cient to guess the theoretical upperbounds on achievable cooling rates. Note that estimates of hbased on boundary-layer theory are applicable only when thesample size L is much larger than the boundary layerthickness, t (or Re L >> 1), which is true in N 2 gas at T = 100 Kfor L >> 20 mm, in He gas at T = 30 K for L >> 100 mm, inliquid N 2 for L >> 0.2 mm, in liquid ethane for L >> 2 mm andin liquid propane for L >> 10 mm.2.2. Characteristic cooling timesTo cool a crystal, heat must be conducted through thecrystal's internal thermal conduction resistance from itsinterior to its surface and then through the convective thermalresistance of the gas or liquid boundary layer. The bottleneckin heat transport is determined by the ratio of the boundarylayer'sheat-transport coef®cient per unit area h ' / t to thesample's thermal conductance per unit area s /L, given by theBiot number (Liakhovski, 1940; Zasadzinski, 1988; Kuzay etal., 2001)Bi ˆ hL= s :Table 3(c) gives typical Bi values for gas-stream and liquidplungecooling, using values of the sample's thermal conductancediscussed in x3 estimated from the data in Table 2(a).For gas-stream cooling with both N 2 and He, Bi

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