Transient heat transfer at fluid flow in a thick-walled pipeline

The work aims to determine the transient temperature distribution of the medium and the pipeline wall using the finite difference method. Time courses of the temperature of the flowing medium and pipeline walls caused by a step change in temperature of the medium at the pipeline inlet, obtained by the numerical method, were compared with the courses calculated using strict analytical formulas. The numerical method of determining the transient distribution of temperature of medium and pipeline wall can be used in the analysis of heating and cooling of heating or steam pipelines with any changes in time of mass flow rate of the flowing medium or temperature of the medium at the inlet to the pipeline..


Introduction
During start-up and shutdown of power units, heating and cooling of pipelines, in particular, the pipeline connecting the boiler with the turbine, is an important issue. The temperature course of the medium is important not only because of the thermal stresses in the pipeline wall but also because of the stresses in the fittings installed on the pipeline [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. In the case of steam or gases, the temperature changes along the pipeline length are higher due to the lower density and specific heat of the medium. Large wall thicknesses and considerable lengths also contribute to a significant reduction in the temperature of superheated steam in pipelines.

Mathematical formulation of the problem
The temperature changes of the medium and the pipeline wall are described by the following equations The boundary condition and initial conditions are in the form cz pcz cz r cz pcz cz cz in cz in (1-2) have been derived with the assumption that the pipeline wall is thermally insulated on the external surface. Also, it has been assumed that the distribution of temperature and fluid velocity in the cross-section of the pipeline does not depend on the radius. Due to the complexity of the problem described by the Eqs. (1)(2)(3)(4)(5), numerical methods, e.g. presented in the paper [17], must be used to determine the transient temperature of the medium and the pipeline. The initialboundary value problem (1)(2)(3)(4)(5) was solved with the explicit finite differences method. Fig. 1 shows a diagram of the differential grid for the calculation of the temperature distribution of the medium and the pipeline. Applying the explicit finite differences method to the solution of the initial boundary problem (1)(2)(3)(4)(5), it is obtained where tn = nt, t -denotes the time step in s, and z z + = z/Lr is dimensionless co-ordinate. After solving the equations (9-10) in respect to i = 1, …, N, n = 0, 1, 2, … In order to ensure stability of calculations using Eqs. (13)(14), the Courant condition [4,5] must be met where wi is the fluid velocity in the i-th node. In a particular case, when f(t) = Tcz the problem described by the Eqs.  , (17) where I0 is the modified Bessel function of the zero order and the U is the function defined by the formula In Eqs. (16-18) the following designations were adopted: Eqs. (16)(17)(18)(19) will be used to assess the accuracy of the finite difference method. In order to obtain a high accuracy of the analytical solution, the double series (18) was taken as n > 400 in the computer calculations, as the series is slowly converging.

Numerical simulation of steam pipeline heating
The steam pipeline to be analysed is made of 13HMF steel. Its dimensions are as follows: length Lr = 48 m, internal diameter din = 0.217 m, and wall thickness gsc = 0.028 m. The physical properties of the pipe material are: density m = 7650 kg/m 3 , specific heat cm = 519 J/(kgK) and thermal conductivity m = 35 W/(mK). It has been assumed that superheated steam with temperature Tcz = 540C and pressure pcz = 11 MPa flows suddenly into the pipeline. The steam thermal properties are -density cz = 28,492 kg/m 3 , specific heat ccz = 2484 J/(kgK) [19].
The initial temperature of the pipeline and fluid is T0 = 300C. The value of heat transfer coefficient on the inner surface of the pipeline, for the preset flow velocity of the fluid wcz, was calculated from the Dittus-Boelter correlation formula. The distribution of fluid temperature Tcz and pipeline wall temperature Tsc as a function of time was determined from the solution of the system of differential Eqs. (13)(14) and analytical Eqs. (16)(17).
The analysis of Figs. 2 and 3 shows that the finite difference method gives results with very good accuracy. It can be noticed that at higher steam speeds both the medium and the wall reach the steady state faster. Finite difference method can be applied for any changes in time of the medium temperature at the inlet and at temperature and pressure dependent thermal properties of medium and wall material. In such cases, it is difficult or impossible to find an analytical solution.

Conclusion
The explicit finite difference method presented in this paper can be used for the analysis of pipeline heating and cooling. A comparison of the calculated time histories of fluid and pipeline wall temperature with a stepwise increase in fluid temperature at the pipeline inlet with the histories obtained using strict analytical formulae shows that the accuracy of the differential method is very good. The finite difference method can be applied to any temporary changes in temperature of the medium at the inlet to the pipeline and depending on the temperature of thermo-physical properties of the medium and wall.