CORRELATION OF HEAT TRANSFER IN A CYLINDER

whereg acceleration due to gravity,P coefficient of expansion of air,AT Trne - Tc,,v kinematic viscosity of air,C Y thermal diffusivity of air.Equation 2.4 is valid for 10” Ra, 10”.The heat transfer coefficient due to natural convection to air is shown as a function of cylindersurface temperature in Fig. 2.1. A cylinder diameter of 4.0ft and a fire temperature of 1475 F wereassumed. Properties of air were taken from Incropera and DeWitt.’ As evident from Fig. 2.1, radiationis the dominant mode of heat transfer to the cylinder exterior during a fire. Radiation heat transfercoefficients are at least an order of magnitude higher than corresponding convection heat transfercoefficients. These results are consistent with observations made by Mansfield. Io Mansfield observedthat the ratio of radiative to convective heat flux is approximately 9:l for large pool fires.The effect of assuming forced convection as opposed to natural convection over the cylinderexterior has been investigated. Based on hydrocarbon pool fire data“, initial vertical velocities ofapproximately 13 ft/s (4 m/s) are achieved in a pool fire. Using this velocity and an appropriatecorrelation for forced convection over a cylindef‘, forced convection heat transfer coefficients werecalculated. The forced convection values were found to be of the same order of magnitude as thosepredicted by natural convection correlations. Therefore, it can be concluded that the convective heattransfer over the cylinder exterior may be accurately predicted by the natural convection correlationpresented earlier in this section.7

3. HEAT TRANSFER FROM THE CYLINDER TO CONTENTSDuring heating of the cylinder, heat transfer occurs between the cylinder and all three phasesof UF, (solid, liquid, and vapor). Prior to liquid overflow, heat transfer to the solid UF, bulk occursacross a vapor gap on the bottom portion of the cylinder. Radiative heat transfer also takes place betweenthe upper cylinder interior and the UF, solid top surface. Heat transfer from the cylinder to liquid UF,in contact with the cylinder occurs by natural convection and/or boiling which will be discussed inChapter 5 . Following liquid overflow, radiative heat exchange occurs between the upper portion of thecylinder and the overflowing liquid surface. Heat transfer to the UF, vapor occurs by natural convectionalong the upper interior of the cylinder.3.1 HEAT TRANSFER TO THE SOLID UF6 BULKHeat transfer to the solid UF, bulk occurs by convection and conduction as well as radiantexchange across an annular gap containing UF, vapor. Kuehn and Goldstein have correlated equationsfor convection and conduction between horizontal circular cylinders. The following relation can be usedto approximate the Nusselt number for heat transfer by natural convection between an inner and outerhorizontal cylinder:whereX [ ( 0.51SRu [ 1 (0.559/Pr)3/5)I5 (ip151/150.1hm ) ],and(3.4)9

RuDiis the Rayleigh number based on the inner diameter (DJ,Ram is the Rayleigh number based on theouter diameter (Do),and Pr represents the Prandtl number. Using this nomenclature, the gap thicknessis defined ast8w-Do- Di2(3.5)All properties in Eqs. 3.1 - 3.4 are evaluated at the film temperature (Tfi,m)calculated aswhereTcy, inner cylinder surface temperature,Tsolid UF6 solid surface temperature.For heat transfer across the annulus due to conduction, Kuehn and Goldstein recommend thefollowing correlation for Nusselt number?The value of the overall Nusselt number for heat transfer across an annular gap (convectionconduction) valid at any Rayleigh number is estimated by (3.8)The corresponding heat transfer coefficient is calculated asHeat is also transferred across the vapor gap by radiative exchange. This radiative heat transferis calculated as10

(3.10)whereAwl surface area of solid UF, exposed to uF6 vapor,F, emissivity factor,Q Stefan-Boltzmann constant,Tql inner cylinder surface temperature (absolute),Twl solid UF, temperature (absolute).The emissivity factor, F,, can be estimated as3F,1 - 1 - cy1whereeq1 esol sol1'(3.11)emissivity of cylinder,emissivity of solid UF,.Equation 3.11 assumes that A,exchange is A,.The corresponding heat transfer coefficient for this radiative(3.12)Figure 3.1 illustrates the relative magnitudes of the heat transfer coefficients due to convectionand conduction across a gap containing UF, vapor as a function of gap width. A film temperature of400"F, the corresponding vapor pressure at 400"F, and an outer diameter of 48 in (i. e., 48X cylinder)were assumed in order to generate the data appearing in Fig. 3.1. As seen from the data, convection heattransfer is the dominant mode of heat transfer across the vapor gap. Even at very small gap widths (1/8in), the convective heat transfer is an order of magnitude greater than the heat transfer due to conductionacross the gap. Figure 3.2 presents the combined heat transfer coefficient (convection and conduction)in an annular region containing UF, vapor as a function of vapor film temperature and pressure. Anouter diameter of 48 in, a gap width of 1/4 in, and a solid UF, temperature of 100 F were used togenerate this data. As seen from Fig. 3.2, the heat transfer across the gap due to convection andconduction is a strong function of pressure. Values of the combined heat transfer coefficient range fromapproximately 1 to 20 Btu/hr-ft'-"F. The radiative heat transfer coefficient across the vapor gap (h5)isalso shown in Fig. 3.2 as a function of film temperature. Values of this heat transfer coefficient rangefrom approximately 1 to 3 Btuhr-ft?-"F.Prior to liquid overflow, radiative heat exchange also occurs between the top cylinder interior andthe solid UF6 surface and may be calculated as11

1004ConvectionLh0.010III1iO-Ib ''3II.II"'546Gap W i d t h ( i n )Heat transfer coefficients (h3and hJ across UF6vapor gap due to convection andconduction.Fig. 3.1./I100JpsiaIPIRadiant h e a t t r a n s f e r c o e f f i c i e n t0.1i100150200250300350400450Vapor F i l m Temperature ( F )Fig. 3.2. Heat transfer coefficients (h3 h, and hJ in an annular gap containing UF, vapor.12

(3.13)whereA, surface area of upper cylinder interior above solid surface exposed to UF, vaporThe corresponding heat transfer coefficient is calculated as(3.14)This heat transfer coefficient is shown as a function of cylinder temperature in Fig 3.3. This radiativeheat transfer coefficients shown as a function of cylinder temperature in Fig. 3.3. Plots are presentedfor UF, solid temperatures of 70 and 147.3"F. Typical values of steel emissivity range fromapproximately 0.07 to 0.81 . 3 For illustrative purposes, a value of 0.6 was arbitrarily chosen to constructthe plots. The solid UF, was assumed to have an emissivity of 0.9.93.2 HEAT TRANSFER TO THE OVERFLOWING UF6 LIQUIDHeat transfer from the cylinder to the UF, liquid in contact with the cylinder wall occurs byboiling which is discussed in Chapter 5. Once the liquid has overflown the solid UF, bulk, heat istransferred to the liquid UF, surface by radiation. This radiative heat transfer is calculated as(3.15)whereT@liquid UF, surface temperature,emissivity of liquid UF,, surface area of liquid UF, exposed to UF, vapor, surface area of upper cylinder interior above liquid surface exposed to UF, vapor. efiq A,A ,The corresponding heat transfer coefficient is calculated as(3.16)13

(3.17)where14

(3.18)z The corresponding heat transfer coefficient is calculated as(3.19)Properties used to calculate the Rayleigh number should be evaluated at the film temperature ETfilm (T,,, T,,)/2]. The natural convection heat transfer coefficient over the upper cylinder interior is presentedgraphically in Fig. 3.4 as a function of the vapor film temperature and pressure. As seen in Fig. 3.4,this natural convective heat transfer coefficient is very sensitive to pressure. Values range fromapproximately 1 to 24 Btu/hr-ft*-"F.{100400 p s i ,S a t u r a t e d Vapor,150200250300Vapor Temperature ( F )350400450Fig. 3.4. Natural convective heat transfer coefficient (hb between the cylinder and UF, vapor.15

4. HEAT TRANSFER BETWEEN URANIUM HEXAFLUORIDE PHASESHeat is transferred by natural convection between the three UF, phases inside the cylinder. Eachheat transfer mechanism is discussed below.4.1 HEAT TRANSFER BETWEEN LIQUID AND SOLID UF, IN GAP REGIONThe heat which is transferred from the UF, liquid to the bulk solid in the annular region iseffectively natural convection over a cylinder. Kuehn and Goldstein recommend the following correlationfor natural convection heat transfer from a horizontal cylinder valid at any Rayleigh and Prandtl number:,Nu,2 2wherewhereRa, Rayleigh number based on diameter,Pr Prandtl number.The corresponding heat transfer coefficient is calculated asNu, kh, -,whereD(4.3)k thermal conductivity of liquid UF,,D diameter of solid UF, lump.Values of the natural convection heat transfer coefficient calculated from the above equations arepresented in Fig. 4.1 as a function of the liquid temperature. A diameter of 4 ft and a solid UF,temperature of 147.3"F were used to generate this data. The effect of pressure on this heat transfercoefficient is negligible. The value of this heat transfer coefficient ranges from approximately 25 to 200Btu/hr-ft'-"F.17

250ALI200I-IsL&IL%IgYI15050100150250200300350400L i q u i d Temperature ( F )Fig. 4.1.Convective heat transfer coefficient (h9)between m6 liquid and solid bulk in gapregion.4.2 HEAT TRANSFER BETWEEN LIQUID W6AND SUBMERGED m6 SOLIDIf heat is continually added to the cylinder, the liquid UF, will eventually overflow the solid.After this occurs, heat is transferred from the UF, liquid to the submerged solid. This heat transfer isapproximated as natural convection over the upper surface of a horizontal flat plate. The Nusselt numberfor natural convection over the upper surface of a cool horizontal plate can be approximated byNu,Equation 4.3 is valid for 10scalculation is defined aswhere 0.27Rai". Ra, 10". The characteristic length, L, used in the Rayleigh numberA, solid surface area exposed to liquid UF,,P perimeter of solid exposed to liquid UF,.18

h,, Nu, kL-.(4.6)In this calculation, the solid surface is approximated as a flat surface. The natural heat transfercoefficient between liquid UF, and the submerged solid bulk is presented graphically in Fig. 4.2 as afunction of the liquid film temperature. A characteristic length of 1.11 ft and a solid UF, temperatureof 147.3"Fwere used to generate this data. The effect of pressure on this liquid convective heat transfercoefficient is negligible. As seen from Fig. 4.2, the values of this heat transfer coefficient range fromapproximately 7.5 to 27.5 Btu/hr-ft2-"F.4.3 HEAT TRANSFER BETWEEN VAPOR UF6 AND THE SOLID BULKHeat is transferred from the UF, vapor to the solid bulk prior to liquid UF, overflow. Thisnatural convective heat transfer may be calculated from Eqs. 4.4 to 4.6. The natural heat transfercoefficient at the vapor-solid interface is shown as a function of the vapor film temperature in Fig. 4.3.A characteristic length of 1.11 ft and a solid UF, temperature of 100 F were used to generate this data.The convective heat transfer coefficient (h,J between the vapor and solid UF, ranges from 0.6 to 8Btu/hr-ft2-"F over the range of temperature and pressure considered.30fJ0I100120." "140'IC.160 .l"'l.200180. I'.220L i q u i d Film Temperature,240.'.I260 .280I.'.300(F)Fig. 4.2. Convective heat transfer coefficient (h,J between UF', liquid and submerged UF,solid.19

100S a t u r a t e d Vapor1O D150200250300350400450Vapor F i l m Temperature ( F )Fig. 4.3. Convective heat transfer coefficient (h,) between UF'6 vapor and solid bulk.4.4 HEAT TRANSFER BETWEEN u F 6 VAPOR AND LIQUID UF',Once the liquid UF, overflows the solid bulk, natural convective heat transfer will occur at thevapor-liquid interface. In this case, the liquid surface is actually flat and therefore the heat transfer maybe calculated by Eq. 4.6. This natural convection heat transfer coefficient (h,J is presented graphicallyas a function of vapor film temperature in Fig. 4.4. A characteristic length of 1 . 1 1 ft and a liquid UF,temperature of 147.3"F were used to generate the data in Fig. 4.4. The convective heat transfer betweenthe UF, vapor and liquid UF, ranges from approximately 0.2 to 8.1 Btu/hr-ft2-"F over the range oftemperature and pressure considered.20

100/S a t u r a t e d Vapor0.1100150200250300Vapor Film T e m p e r a t u r e ( F )350400450Fig. 4.4. Convective heat transfer coefficient (h,) between UF, vapor and liquid UF,.21

\

5. URANIUM HEXAFLl ORIDE POOL BOILINGHeating of UF, storage cylinders at pressures greater than or equal to 22.046 psia will lead toboiling of UF, liquid. Therefore, as a part of the thermal analysis of the UF, cylinders, correlations forthe boiling heat transfer coefficients must be obtained for each possible boiling regime. Although boilingdata for common substances such as water and hydrocarbons are abundant in the literature, such data arenonexistent for UF,. Therefore, a boiling curve for UF, must be constructed from existing boilingcorrelations found in the literature.Figure 5.1 represents a typical pool boiling curve. actual boiling data would result in a smooth,more rounded plot than the one shown in Fig. 5.1; however, the segments of the boiling curve betweenpoints A, B, C, and D are very nearly linear on a log-log plot. Correlations for the various boilingregimes (free convection, nucleate, transition, and film boiling) are discussed in Sections 5.1 to 5.4. Theeffects of surface condition, pressure, and liquid subcooling are discussed in Sections 5.5 to 5.7. Thecomplete set of boiling correlations is then summarized in Section 5.8.5.1 FREE CONVECTION BOILINGThe excess temperature, AT,, is defined as(5.1)Critlcal Heat FluxBD*wNucleateBoiling900BurnoutFilm Boiling.-(Lcidcniroa PolatNucleab BoilingConvectionlog (Twall-Tsat)Fig. 5.1. Typical b o i i curve: surface heat flux as a function of excess temperature.23

whereTwa,, cylinder wall temperature,T, saturation temperature.For small values of AT,, the heat transfer is controlled by natural convection effects. It is assumed thatthe majority of liquid UF, boiling in the cylinder will occur in an annular region between the cylindershell and the solid UF, bulk. Some liquid UF, boiling will occur above the solid bulk. However, thenatural convection heat transfer coefficients for this situation are not expected to vary significantlyfromthose predicted for the annulus. Correlations for natural convection heat transfer between horizontalcircular cylinders (Le., annulus) are presented in a report by Kuehn and Goldstein (see Section 3.1).6Data on UF, obtained from these equations were correlated to obtain the following expressions for naturalconvection heat transfer between the inside cylinder wall and liquid UF, at various pressures:0.3161,p 25 psiahNC 22.596ATep 50 p i ahNc 22.299ATe0.31% ,p loopsia0.3228hNc 21.688ATe,250 psiahNc 20.910ATe0.3282 ,p 400 psiahNc 20.796ATe0.3308 ,p where h, , the natural convective heat transfer coefficient, is in Btu/hr-ft'-"F. Equations 5.2 - 5.6 werecombined to obtain the following expression for natural convection boiling of UF, as a function of excesstemperature and pressure, p , in psia:bhNc aAT, ,(5.7)wherea 22.878 - 1 . 2 9 6 1 0 - p 1 . 9 4 9 1 0 - ,p andb 0.3151 8 . 1 8 9 1 0 - -p 1.075 10-'p .24(5.8)

The corresponding heat flux may be determined byI1qMc AT: .(5.10)It should be noted that the natural convective regime is affected by liquid subcooling (i.e., liquidbelow the saturation temperature). The effect of subcooling on the position of the boiling curve appearsto depend on the convection geometry.8 This effect was not considered in the present study.5.2 NUCLEATE BOILINGAt a certain value of AT,, termed the onset of nucleate boiling, vapor bubbles will begin to formand the heat transfer coefficient will rise rapidly with increasing AT,. Nucleate boiling continues untila maximum value of heat flux, known as the critical heat flux (Point B in Fig. 5. l), is reached. At thislatter point, a considerable amount of vapor is being formed and liquid is no longer continuously wettingthe surface. The most widely accepted and used correlation for nucleate boiling was developed byRohsenow’(5.11)whereplh,gg,p1pv(T liquid dynamic viscosity, conversion factor,liquid density,vapor density,surface tension,liquid specific heat,Rohsenow boiling constant,liquid Prandtl number. heat of vaporization, acceleration due to gravity, cp1 CS f Prl Equation 5.11 is of the form(5.12)where25

3(5.13)All properties in the preceding equation are evaluated at the saturation temperature of the liquid. Thevalue of the constant CSfdepends on the surface-liquid combination. Representative values of theRohsenow constant range from 0.0025 to 0.1 High values of Csf imply smooth, polished surfaceswhile lower values are characteristic of more rough, pitted surfaces.The value of excess temperature at boiling incipience, AT,,(Point A in Fig 5.1), can bedetermined by setting Equations 5.10 and 5.12 equal to one another and solving for AT,. The result is.’(5.14)Zuber obtained an expression for the critical heat flux of the form’(5.15)Replacing the Zuber constant ( 124)by an experimental value of 0.149 and approximating the last termwith unity, Equation 5.15 becomes’(5.16)All properties in Eq. 5.16 are evaluated at the liquid saturation temperature. The temperature differenceat the critical heat flux can be calculated from the following expression:(5.17)26

5.3 TRANSITION BOILINGAs the excess temperature increases beyond the value associated with the critical heat flux, theboiling heat transfer coeffi

3 . heat transfer from the cylinder to contents . 9 3.1 heat transfer to the solid uf, bulk . 9 3.2 heat transfer to the overflowing uf, liquid . 13 3.3 heat transfer to the uf, vapor . 14 4 . heat trans