Este sitio usa cookies para mejorar su experiencia de uso del sitio web.
Si continua navegando, usted acepta el uso de nuestra política de uso.

Scope

About the scope of the program

The developed program has as its starting point the so-called Graetz problem. Such a problem describes the behavior of a Newtonian fluid circulating through parallel plates or in a circular duct, in both cases it is a flow with axial symmetry, starting from a certain point of the axis, where it is fixed as x=0, a sudden change in temperature on its surface. The flow is considered to be fully developed, therefore, the velocity profile is known and the boundary condition, either temperature or heat flow, is also known, the axial conduction condition in the fluid is taken into account, which which makes the Péclet Number (Pe) play a relevant role.

Cases and limitations

Presented the equations that govern the problem and the boundary conditions, it remains to establish the scope of the program. The program offers the field of temperatures both in the fluid and in the duct, under the conditions described above. It is not a design and calculation program for ducts, nor are the surface conditions of the ducts, coefficient of friction, primary or secondary load losses taken into account.

The program considers an invariable parabolic velocity profile for the fluid along the entire length of the tube, as well as constant thermal coefficients. In later developments of the program these limitations will be overcome.

Physical-mathematical model

Graetz's conjugate-extended problem

This problem has been analyzed by numerous researchers, L. Graetz (1883) and Nusselt (1910) were the first to approach solutions to this problem, later, other authors, over the last few years, have proposed solutions analytical and numerical depending on the geometry and the boundary conditions that are applied. A detailed analysis of this problem and its variants can be found in reference [1], in reference [2] the Graetz problem is analyzed in its classic version following the work of Bilir [1], but applying for its resolution the numerical technique of the network simulation method, a numerical method based on finite differences that consists of establishing a relationship between electrical components and the discrete terms of the physico-mathematical equations that model the problem, as will be detailed later.

In the classical formulation of this problem, the condition of axial symmetry is fundamental, which allows the flow to be treated in a simplified way, using 1D models, with which the problem is reduced to the solution of a single nonlinear differential equation . The value of the Péclet number that accompanies the conduction term in the fluid along the axis is important in modeling the problem, for low Péclet numbers the axial conduction term is relevant, as the value of this parameter increases, the term of driving to which it accompanies ceases to have a relevant role, being able to disregard this term and achieving a simpler model.

Hypothesis

In what follows, the study is limited to the problem in circular ducts and the hypotheses and equations that govern the described problem are described, they are the following:

  • Steady state for the velocity field and temperature field and fully developed laminar flow.
  • Constant fluid physical properties, which implies incompressible fluid.
  • Field of velocities and temperatures in 3D.
  • Temperature field varies with angle.
  • There is no viscous dissipation.
  • The heat transfer in the tube wall is taken into account.
  • In addition, a boundary condition is added that breaks the axial symmetry, the lower part of the tube is isolated, it is adiabatic.
  • Buoyant forces are not considered, the Boussinesq approximation is not taken into account.
  • The heat transfer in the tube through which the fluid circulates is considered

Equations

With the conditions described above, the system of equations to be solved is as follows:

This first equation corresponds to the 3D conduction equation in the duct, the first term of the equation follows Fourier's Law for conduction in cylindrical coordinates and the second term refers to the accumulation of heat in the solid, if you take it into account,

In the second equation it corresponds to the energy balance in a differential element of the fluid, being the first term the conduction of heat in the fluid and the second term encompasses the accumulation of heat and the transport of energy in the axial direction, this term is influenced by the Péclet Number;

Boundary conditions

The boundary conditions are:

The first boundary condition refers to the temperature field of the fluid, it is assumed that the entire fluid is at a uniform temperature T1, the second boundary condition refers to the transfer of radial heat at infinity, which is null, the following condition refers to the continuity of heat transfer in the solid-fluid interface, in the following condition the temperature T2 is represented and the point at which this temperature change from T1 to T2 occurs, which affects the upper part of the tube, the last condition refers to the adiabatic condition in the lower half of the tube.

Numerical method

The program uses the Network Simulation Method (MESIR) for its resolution.

Once the system of equations that governs the problem and the corresponding boundary conditions have been established, the network simulation method is applied to solve the problem. The network simulation method consists of establishing a correspondence between electrical elements and the terms resulting from discretizing the equations that govern a problem using the finite difference method. From what has been said above, the following discretized equations can be deduced by means of finite differences from the definition equations of the physical process of study.

Once the equations that govern the problem have been discretized, the correspondence between the elements of the equations and electrical components is established following the well-known analogy between heat and electrical components. Following this analogy, the temperature can be related to the potential difference and the heat to the current flows, in this way, each of the terms of the previous equations can be related to a current flow and applying Kirchoff's law of streams you can put:

The resulting electrical diagram can be seen in the following figure:

About the MESIR

The network simulation method, MESIR, is a numerical method widely used in solving problems in different fields of physics and engineering. The MESIR network simulation method is based on making network models made up of electrical components whose equations are formally equivalent. to those obtained after the spatial discretization of the mathematical models through finite differences. Time is a continuous variable in this model. The volume on which the heat transfer equations are applied corresponds to the one described in the Figure of the electrical diagram. The composite fin has been divided into three different regions. In each of these regions, the heat conduction equation has been applied under 2-D models and their corresponding boundary conditions, the same figure also represents the cell or elementary volume of dimensions, ºz and ºy, resulting from this process.

References

  1. Bilir, S., 1992. Numerical-solution of Graetz Problem with axial conduction. Numerical Heat Transfer Part a-Applications 21, 493–500.
  2. Zueco, J., Alhama, F., Fernandez, C., 2004. Analysis of laminar forced convection with Network Simulation in thermal entrance region of ducts. Int. J. Therm. Sci. 43,443–451.
  3. Seco-Nicolás, Manuel; Alarcón García, Mariano; Alhama, Francisco. 2018. Thermal behavior of fluid within pipes based on discriminated dimensional analysis. An improved approach to universal curves. Applied Thermal Engineering. DOI: 10.1016/j.applthermaleng.2017.11.091
  4. Seco-Nicolás, Manuel; Alarcón García, Mariano; Luna-Abad, Juan Pedro., 2020. Experimental Calculation of the mean temperature of flat plate thermal solar collectors. Results in Engineering 5. DOI: 10.1016/j.rineng.2020.100095.
  5. Seco-Nicolás, Manuel; Alarcón García, Mariano; Luna-Abad, Juan Pedro., 2021. 3D numerical simulation of laminar forced-convection flow subjected to asymmetric thermal conditions. An application to solar thermal collectors. Solar Energy. DOI: 10.1016/j.solener.2021.02.022
  6. Mariano Alarcón, Fernando Manuel Martínez-García, Félix Cesáreo Gómez de León Hijes, Energy and maintenance management systems in the context of industry 4.0. Implementation in a real case, Renewable and Sustainable Energy Reviews, Volume 142, 2021, 110841. DOI: 10.1016/j.rser.2021.110841.
  7. Alarcón García, Mariano;Seco-Nicolás, Manuel; Luna-Abad, Juan Pedro; Ramalo-Gonzalez, Alfonso P. 2022. Forced laminar flow in pipes subjected to asymmetric external conditions: The HEATT© Platform for online simulations. Intechopen. DOI: 10.5772/intechopen.107215