Technical Assessment of Dry Ice Limits on Aircraft (2013)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

19 where h is the heat transfer coefficient, J/cm2 ? s ? °C, L is the characteristic length of the surface, cm, and k is the thermal conductivity, kJ/m ? s ? °C. Grashof Number = L Tg3 2 2 ρ β µ ∆ where b is the thermal expansion coefficient, 1/°C, DT is the temperature difference between the surface and the bulk fluid, °C, and g is the acceleration due to gravity = 9.8 m/s2. Because the Grashof and Prandtl numbers are often used together, the Rayleigh number is defined by: Ra Gr= Pr This latter parameter is used to estimate the value of h for the outer surface of the package. When estimated in this way, h is proportional to the product of the Grashof and Prandtl numbers taken to the ¼ power. Probably more important for sublimation is the dimen- sionless number that is formed by expressing the sublimation rate of the dry ice as a function of the rate of heat transfer through the insulation and rearranging the equation to make the resulting equation dimensionless: Sublimation Number = = kL T t r T R rs s a 2∆ ∆ λ λ where l is heat of sublimation, kJ/kg, t is the thickness of the insulation, m, r is the sublimation rate, kg/s, L2 is the total surface area of the package, m2, and DT is the temperature difference between the sublimation temperature and the ambient temperature, °C. Dimensional Analysis for Heat Transfer Even without doing experiments or heat transfer calcu- lations, dimensional analysis can provide insight into the variables that govern carbon dioxide production. First, it is important to realize that for a given amount of dry ice, the sublimation rate* is entirely determined by the rate of heat gain. Once heat reaches the dry ice, the dry ice sublimes and the gaseous carbon dioxide leaves quickly through a bulk flow of gas. (A more detailed discussion of the importance of heat transfer is presented in Chapter 7.) The equations governing steady-state heat transfer are frequently expressed in terms of dimensionless numbers. For the analysis of heat transfer from the air to the outer sur- faces of the packaging, there are three major dimensionless numbers: the Prandtl, Nusselt, and Grashof numbers. The Nusselt number governs convective heat transfer, and the product of the Grashof and Prandtl numbers, also called the Rayleigh number, governs buoyant convective heat transfer. The definitions are: Prandtl Number = C k pµ where Cp is the specific heat, kJ/kg ? °C, µ is the kinematic viscosity, N?s/m2 or kg/m ? s, and k is the thermal conductivity, kJ/m ? s ? °C. Nusselt Number = hL k C h a p t e r 6 Dimensional Analysis of Dry Ice Sublimation *By sublimation rate is meant the mass of dry ice per unit time that changes from a solid to a gas. In SI units, this would be grams or kilo- grams of dry ice per hour.

20 In the equation, if r is expressed in terms of kg/hr, then the conversion factor of 3,600 s/hr must be included to keep the value of the sublimation number dimensionless. The subli- mation number can be modified slightly by combining the rate and area term into an area-normalized loss rate term. That modification is also shown in the second expression, where ra is an area-normalized loss rate (kg/m2 ? hr). In addi- tion, since t/k defines the R number for the insulation, the sublimation number can be simplified even more. Since the temperature difference and the amount of energy required to sublime a kilogram of dry ice are constants, the only terms that can be controlled are the dimensional terms and the recipro- cal of the thermal conductivity divided by the thickness of the insulating barrier—the R value of the insulation. Dimensional Analysis for Ventilation Flow There is another part of the dimensional analyses that is related to the flow of air through a compartment. That equa- tion is first expressed with dimensional terms, and then terms are combined to make the equation dimensionless. First, the dimensional case: C F E N fr L E N C F C Fin in CO p p a c c out out out inρ 2 2+ + + = = ρCO2 In the equation, F is the volumetric flow rates in and out, and since the airplane pressure is maintained at a constant, the volume flow in must equal the flow out, the volume associated with the generation of carbon dioxide being neglected. The second term is the emission rate from a passenger, Ep, times the number of passengers, Np. Then we have the emission rate from the dry ice packages expressed as ra times their total area L2 times the fraction of in leakage from the cargo compart- ment f. The final term is the emission rate from any carts using dry ice, Ec, times the number of such carts, Nc. For the purposes of dimensional analysis, these three terms can be combined into one term, the emission rate in the passenger cabin, Rc. Thus, in dimensionless terms, the two terms are: Cout/Cin and Rc/CinFinrCO2 as shown in the following equation: R C F C C c in in CO out inρ 2 1= − Since Cin is currently equal to a volume or mole fraction of 0.00039 or 390 ppm, if Cout is the FAA regulatory limit of 5,000 ppm (mole fraction 0.005), then the production rate of carbon dioxide in the main cabin divided by the rate of carbon dioxide entering into the ventilation system with the outside air, Rc/CinFinrCO2, cannot be greater than 11.8. This equation can be rearranged to express the limit in terms of air exchanges per hour. The resultant equation is: R C F R C ACH V c in in CO c in comp COρ ρ2 2 11 8= = . . The second expression expresses the inlet volumetric flow in terms of the air exchanges per hour (ACH) and the volume of the compartment, Vcomp. Once again, note that the mass of dry ice in the aircraft does not appear in either form of this equation. The FAA CO2 concentration limit of 5,000 ppm is for areas occupied by passengers and crew. When packages containing dry ice are shipped, most are placed in the cargo compartment areas of the airplane. Some air carriers use a 30,000-ppm CO2 limit for cargo compartments containing live animals. If this limit were used for ventilated cargo compartments containing dry ice packages, then the equation becomes: R C F R C ACH V c in in CO c in comp COρ ρ2 2 75 9= = . . Using this higher limit for areas not used by passengers or crew results in an allowed production rate of CO2 that is more than six times the total CO2 production rate allowed in the passenger cabin. Discussion of Dimensional Analysis Results It is believed that the sublimation number can be used to describe any carbon dioxide release rate from packages. If this is the case, then parameters such as thermal conductivity of the package, the area available for heat transfer, and the dif- ference between the temperature inside the package and the outside ambient temperature are all important variables; the mass of dry ice does not appear in any of the equations. On the other hand, the equations do contain an area term that is a measure of the size of the package. Using a sublimation rate based on the mass of dry ice in a package is not supported by the dimensional analysis and does not make technical sense. It is recognized that the mass of dry ice has been used for setting aircraft limits for decades. While it is in none of the equations, why might it still be an effective limit? One observa- tion is that most packages are loaded so that they can maintain the desired internal temperature for several days, and such a thermal performance rate is easily attained with standard EPS insulation. The generous use of EPS is the underlying assump- tion behind much of the dry ice loading—using this much insulation can provide a large factor of safety, making the use of mass loss rates acceptable.