Generalized Theory of Convective Heat Transfer in a Free-Molecule Flow

Abstract

The theoretical analysis of convective heat transfer in a freemolecule flow with Maxwellian velocity distribution is generalized by reducing all relationships to a form common to all gases irrespective of their molecular structure. A simplified computingprocedure is set up, and the heat-transfer characteristics of a flat plate, cylinder, and sphere are presented and correlated graphically. A method of applying these results to a cone and to compound shapes is described. NOMENCLATURE cp Cv D E e erfs erfcs F = specific heat capacity at constant pressure, (ft.lbs.)/(lb.sec./ft.)°F„ = specific heat capacity at constant volume, (ft.lbs.)/(lb.sec./ft.)°F. = reference diameter, (ft.) = molecular energy transport rate, (ft.lbs./sec.) = molecular energy transport rate per unit surface area, (ft.lbs./ft.sec.) = base of natural logarithms 2.7183 error function ' dx, (dimensionless) = complementary error functions, 1 — erfs, (dimensionless) = average surface integral defined by Eq. (2.13), if/ erage s h his) f dS, (see Q) = average surface integral defined by Eq. (2.14), : dS, (see Q) -Taw), = heat-transfer coefficient, Q/(TW (ft.lbs./ft.sec. °F.) = modified Bessel function of first kind and zeroth order, / ^ ^ o s ^ d , (dimensionless) Ii(s) = modified Bessel function of first kind and first order, -s cos 0 c o s ^ ^ (dimensionless) ierfcs = integral of the complementary error function, erf ex dx, (dimensionless) Jo i: J k M N Nu = number of degrees of freedom, (dimensionless) = Boltzman constant 5.66.10~ 4 (ft.lb./molecule°F.) = Mach Number (dimensionless) = number of molecules per unit volume (molecules/ft.) = Nusset Number JID/K (dimensionless) Received May 21, 1952. * This work was done as part of an Office of Naval Research contract with the Department of Engineering, University of California at Berkeley. f Assistant Professor of Mechanical Engineering. Pr Q