Technical Note Flow characteristics in a heated rotating straight pipe

Abstract

Abstract The flow in a rotating or in a heated straight pipe has been extensively studied not only for academic interest, but also for the great importance in mechanical applications such as in pipe heat exchangers, in cooling systems of rotor blades in gas turbines and in chemical mixing. The viscous flow in a straight pipe rotating about an axis perpendicular to its own, has as a result the generation of a secondary flow that is sustained by the Coriolis force introduced by the rotation of the pipe. Barua [1] used a regular perturbation about the Poiseuille flow limit, similar to Deans [2, 3] approach for stationary and curved pipe flow. He showed that rotation generates a secondary flow and that it depends on the non-dimensional parameter R r = (2Ω α 2 ⧹ ν ), where Ω is the angular frequency of rotation, α is the radius of the pipe and ν is the kinematic viscosity. Subsequent boundary-layer analysis also predicted a significant increase in the friction factor with rotational speed for small rotational rates and high axial pressure gradients (Mori and Nakayama [4] , Ito and Nanbu [5] ), the latter group also obtaining satisfactory agreement with their experimental results. Mansour [6] considered higher rotational velocities using a computer extension for the perturbation expansion, similar to a method that was applied by Van Dyke [7–9] who studied the flow in a stationary straight pipe. According to this method the equations of motion are modified, so that they are depend on a single parameter K = R e R r , under the assumption that R e → ∞, R r → 0, where R e is the Reynolds number based on the axial velocity. Benton [10] considered small rotational velocities and constructed a small perturbation expansion about the Hagen–Poisseuille flow. Later, Benton and Boyer [11] assumed the case of a rapid rotating conduit with R r R e ≤ 1. Duck [12] used a numerical procedure, based on a combination of Fourier decomposition and finite difference discretization to study the flow structure in rotating circular ducts. The first experiments concerning a rotating pipe were conducted by Trefethen [13] who observed that rotation transfers the onset of turbulence to higher Reynolds numbers. Later, Euteneuer and Piesche [14] in their experimental studies in circular pipes confirmed that the pressure drop is significantly higher than that for straight pipes, in agreement with the theoretical results. The earliest analysis on the flow in a heated straight pipe was considered by Morton [15] . His study was restricted to small rates of heating and he obtained solutions for the axial velocity and temperature as power series depending on the parameters R e R a , where R a is the Rayleigh number based on the temperature gradient along the pipe wall. Mori and Nakayama [16] assumed velocity and temperature boundary layers along the pipe wall and analysed theoretically the flow field and the temperature field. Van Dyke [17] modified Mortons variable in order to clarify the dependence of the problem on the parameters P r , R a and R e , where P r is the Prandtl number. The advantage of these simplifications was that the flow depended only on two parameters e = P r R a R e and P r , respectively. Guiasu et al. [18] were able to compute many terms of the series on Mortons problem using symbolic computation packages. In the present work we study the fully developed steady flow in a straight rotating heated pipe with circular cross-section. The equations of motion and energy depend on three parameters that characterize the flow, the rotational Reynolds number R r , the Reynolds number based on the axial flow R e and the Rayleigh number R a , and they are solved both analytically and numerically. In the analytical solution the functions of the flow are expanded in power series of the parameters R r and R a . Because of the difficulties of the problem introduced by the presence of these three parameters, we were able to compute only ten terms in each series, so that the range of values of the parameters for which the analytical solution converges is limited by the products R e R a R e R r R e R a and R e R r for which the numerical solution is valid are R e R a R e R r