Heat or mass transfer in a fluid in laminar flow in a circular or flat conduit

Abstract

Accurate solutions to the Graetz equation and to the similar equation for flow between two parallel plates are presented including the first ten or eleven eigenvalues and important derivatives. The first six eigenfunctions are also presented at intervals of 0.05 from y = 0 to y = 1. For one directional flow in a flat duct (between two parallel plates) the corresponding equation isThe corresponding equations for mass transfer by diffusion are usually written with concentration or partial pressure in place of temperature, and dsusivity in place of thermal diffusivity.The first published solution to Equation (3) was presented by Graetz (13, 14). The assumptions and boundary conditions made by Graetz are constant thermal diffusivity, constant tube wall temperature, temperature symmetrical about the axis, uniform temperature at tube inlet, Wly developed parabolic velocity profile at tube inlet, and negligible conduction in direction of flow.These assumptions correspond to the following boundary conditions:

Fully developed forced convection of the Phan-Thien–Tanner fluid in ducts with a constant wall temperature

“…(18) requires transformation of variables followed by such techniques as separation of variables or L e evê eque analysis, and was obtained by Brinkman [18] and Ou and Cheng [19], amongst others (see also [14,Chapter 5] and [20]). For the SPTT fluid, the solution of Eq.…”

Thermal modelling of new Li-ion cell design modifications

“…Since this general solution of the Graetz problem is valid for both laminar and turbulent flow, the same applies to the expression for the heat transfer coefficient now to be derived. Eigenvalues determined by Brown [26] are used for laminar flow, and those from Notter and Sleicher [25] for turbulent flow. As the fluid flows through the tube, the wall temperature changes at position ξ .…”

An eigenvalue problem arising in mass and heat transfer studies

“…The earliest work, which dates back to 1883, seems to be that of Graetz [2], who considered the case in which the second boundary condition of (1) is #"(1) = 1. Most recently, Brown [3] recalculated the solutions to Graetz's problem by numerical techniques using a digital computer. However, no solutions pertinent to the present boundary conditions are available.…”