Analytical solution for transient laminar fully developed free convection in vertical channels

Using Green's function method, analytical solutions for transient fully developed natural convection in open-ended vertical circular and two-parallel-plate channels are presented. Different fundamental boundary conditions for these two configurations have been investigated and the corresponding fundamental solutions are obtained. These fundamental solutions may be used to obtain solutions satisfying more general thermal boundary conditions. In terms of the obtained unsteady temperature and velocity profiles, the transient volumetric flow rate, mixing cup emperature and local nusselt number are estimated.Zusammenfassung Für oben und unten offene vertikale Kanäle mit Kreisquerschnitt bzw. als Parallelplattenanordnung werden unter Verwendung der Methode der Greenschen Funktionen analytische Lösungen für die nichtstationäre, vollausgebildete, natürliche Konvektion gefunden und zwar unter Zugrundelegung verschiedener Fundamental-Randbedingungen bezüglich beider Konfigurationen. Die so ermittelten Fundamentallösungen können zur Gewinnung von Lösungen für allgemeine Randbedingungen dienen. Der zeitlich veränderliche Volumenstrom, die Mischtemperatur und die Nusselt-Zahl werden mit Bezug auf die erhaltenen nichtstationären Profile für Temperatur und Geschwindigkeit näher analysiert.

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Analytische Lösung für die nichtstationäre vollausgebildete laminare freie Konvektion invertikalen Kanälen

Nomenclature a local heat transfer coefficient based on the area of the heat transfer surface,q/(T_w–T₀)=±(T/y)_w/(T_w–T₀), minus and plus signs apply respectively for heating and cooling in case of parallel-plate channel and vice versa in case of a tube - average heat transfer coefficient over the channel - c_p specific heat of fluid at constant pressure - f volumetric flow rate, for circular channels and or two-parallel-plate channels - F dimensionless volumetric flow rate,f/(2lvGr^*) for circular channels forfw/(lvGr^*) for two-parallel-plate channels - g gravitational body force per unit mass (acceleration) - G Green's function - Gr Grashof number,±g(T_w–T₀)w³/v² in case of an isothermal boundary of±gqw⁴/2kv² in case of a uniform heat flux (UHF) on the heat transfer boundary, the plus and minus signs apply to upward (heating) and downward (cooling) flows, respectively. ThusGr is a positive number in both cases. - Gr^* modified Grashof number,wGr/l - h heat gained or lost by fluid from the entrance up to a particular elevation in the channel, ₀fc_p(T_m–T₀) for all cases - J₀ Bessel function of zero order - k thermal conductivity of fluid - l height of channel - L dimensionless height of channel,1/Gr^* - Nu local Nusselt number,|a| w/k - average Nusselt number, - p pressure of fluid inside the channel at any cross-section - p pressure defect at any point,p–p_s - p₀ pressure of fluid at the channel entrance - p_s hydrostatic pressure, ₀gz where the minus and plus signs are for upward (heating) and downward (cooling) flows, respectively - p dimensionless pressure defect at any point(pw⁴)/(₀l²²Gr²) - Pr Prandtl number,c_p/k - q heat flux at the heat transfer surface,q=±k(T/y)_w where the minus and plus signs are, respectively, for cooling and heating in case of circular pipe and vice versa in case of a parallel-plate channel - Ra Rayleigh number,GrPr - Ra^* modified Rayleigh number,Gr^*Pr - t time - T fluid temperature at any point - T_m mixing-cup (mixed-mean) temperature over any cross section, for circular channels, and for two-parallelplate channels - T₀ initial and channel-inlet fluid temperature - T_w temperature of the heat-transfer wall - u axial velocity component at any point - U dimensionless axial velocity,uw²/(lvGr^*) - w radius of circular tube or width (between plates) of parallel-plate channel - y radial or transverse coordinate - y dimensionless radial or transverse coordinate,y/w - z axial coordinate - Z dimensional axial coordinate,z/(lGr^*)Greek symbols constant appears in Eq. (8) - parameter appears in Eq. (9) which equals the integration of with respect to or volumetric coefficient of thermal expansion - _n eigenvalues - parameter appears in Eq. (7) - _n eigenvalues - parameter appears in Eq. (12) - _n eigenvalues - parameter appears in Eq. (9) - dimensionless temperature,(T–T₀)/(T_w–T₀) in case of an isothermal heat transfer boundary and(T–T₀)/(qw/2k) for UHF boundary - _m dimensionless mixing cup temperature,(T_m–T₀)/(T_w–T₀) in case of an isothermal heat transfer boundary and(T_m–T₀)/(qw/2k) for UHF boundary - _w dimensionless temperature of the heat-transfer wall, equals unity in case of an isothermal heat transfer boundary and(T_w–T₀)/(qw/2k) for a UHF boundary - _n eigenvalues - dynamic viscosity of fluid - kinematic viscosity of fluid, /₀ - fluid density at temperatureT,₀[1–(T–T₀)] - ₀ fluid density atT₀ - demensionless time,tk/(cw²)