MATHEMATICAL MODELING OF A COMBINED HOT-WATER HEATING SYSTEM BY MEANS OF THE FINITE ELEMENT METHOD

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1 2005/2 PAGES RECEIVED ACCEPTED J. LOVÁS, K. MIKULA MATHEMATICAL MODELING OF A COMBINED HOT-WATER HEATING SYSTEM BY MEANS OF THE FINITE ELEMENT METHOD ABSTRACT Ing. Jozef Lovás Slovak University of Technology - Faculty of Civil Engineering, Department of Building Services (TZB), Radlinského 11, Bratislava Research field: Combined heating, Floor heating Doc. RNDr. Karol Mikula, PhD. Slovak University of Technology - Faculty of Civil Engineering, Department of Mathematics, Radlinského 11, Bratislava Research field: Finite elements method KEY WORDS The subject of this article is an analysis of heat and mass transfer by means of a mathematical model of combined hot-water heating as a combination of 2 heating systems with different methods of heat transfer into an indoor environment. The model combines floor hot-water heating and convection-type heating with radiators. This combination joins the advantages of both systems and eliminates their setbacks, which appear if installed as individual heating systems. Combined heating Floor heating convection heating finite elements method MATHEMATICAL MODEL OF COMBINED HEATING ρ - density (kg.m -3 ), τ - time (sec), The mathematical model is based on a vertical cross section of a room with dimensions of 5 x 5 x 2.8 m with a heating corpus installed on a peripheral wall under a window. The calculation model uses the vertical cross section of the room with a depth of 1m (see the graphic representation in Fig. 1). P - pressure (Pa), (2) MATHEMATICAL FORMULATION For the mathematical solution of the selected model we need to solve the following equations: Equation of Continuity (1) v x, v y, - velocity components in direction of the x, y axes (m/sec), R gas constant (J.kg -1.K -1 ), t temperature (K), Relationship between deformation stress tensor and velocity tensor in the Newtonian fluid (3) (4) 2005 SLOVAK UNIVERSITY OF TECHNOLOGY 15 mikula_01.indd :15:40

2 Fig.1 Computational model of the combined heating system σ stress tensor, u i components of the velocity vector (u 1 = v x, u 2 = v y ) (m.s -1 ), µ - dynamic viscosity (Pa.s), Fluid movement equations with included viscosity Navier-Stokes equations g x, g y - components of gravity acceleration (m.s -2 ) µ e - effective viscosity for a turbulent model (Pa.s) (6) (5) (7) Tab. 1 Boundary conditions for the combined hot-water heating model Boundary condition for: Values Radiator, area O2 t VT1 = 30.5 C, t VT2 = 27.5 C, t VT3 = 24.5 C, ε = 0.7 Edge: H6 t P1 = 21.2 C, t P2 = 21.9 C, t P3 = 21.7 C, ε = 0.93 Edges: H1, H3 t = 15.8 C, ε = 0.93 Edge: H2 t = 4.8 C, ε = 0.92 Symmetry Edges: H4, H5, ε = 0.92 t - temperature ( C), ε - emission ability (-), ν - normal vector 16 mikula_01.indd :15:42

3 Fig.2 Temperature profile in a vertical PDL / VT = 30% / 70% Fig.3 Temperature profile in a vertical PDL / VT = 50% / 50% Fig.4 Temperature profile in a vertical PDL / VT = 70% / 30% (8) t - static temperature ( C), v - absolute value of fluid velocity vector (m/sec), Energy Equation (11) C p - specific heat (J.kg 1.K 1 ), t o - total average temperature (input value) ( C), λ - heat conductivity (W.m -1.K -1 ), (9) (12) (13) (10) BOUNDARY CONDITIONS For the complete mathematical formulation, it is necessary to 17 mikula_01.indd :15:45

4 Fig.5 Horizontal temperature profiles in the room with combined heating at different ratios of the floor heating output power (PDL) to the radiator heating output power (VT) define the boundary conditions. The boundary conditions for the mathematical modeling are determined by the geometry of the heated room where the radiator is located, the prescribed temperature of the heat source, the convection and radiation mechanism on the radiator surface, and the symmetry edges. Inside the heated room the material properties of air (taken from Air-SI (Ansys)) database are considered. RESULTS The following graphs in Fig. 2 to 4 show the temperature profile in a vertical direction at different distances V from the facades for the combined heating system by various combinations of the output power of the floor heating surface and the radiator. Fig. 5 documents the temperature profiles in a horizontal direction, at various distances H from the floor and for different output powers of the hot-water floor heating and convection heating. ANALYSIS OF THE COMPUTED RESULTS From the graphs of the temperature gradient in Fig. 2 to 4, the collaborative effect of the floor heating surface (PDL) and the radiator (VT) on the isothermal curves, especially in the upper part of the radiator, in the vicinity of the window structure, is evident. With the growing temperature of the radiator, the isotherms are turning toward the y- axis. The effect of the heat convection and the induced air flow is the greatest in the vicinity of the radiator, as can be seen from Fig. 2, at a distance of 0.5 m from the peripheral wall. As implied by the above-mentioned figures, the temperature profiles in the vertical and horizontal directions for the combined heating system approximate the ideal type of heating, with the decreasing output power of the radiator. The 70/30% ratio in favor of the floor heating output power is optimal as regards thermal comfort and also for the elimination of the sudden heat loss/gain changes in the interim seasons of the year. The following figures show a comparison of the vertical temperature 18 mikula_01.indd :15:47

5 Fig.6 Vertical temperature profiles in a room with the combined heating system, at different distances from the peripheral wall, for different ratios of output heating power of floor heating surface and of the radiators. profiles for different output power ratios of the floor heating and radiators at different distances V from the peripheral wall. CONCLUSION The results of the theoretical calculation of the combined hot-water heating model with the radiator installed on the cooled external building wall and with the floor heating surface document the heat dissipation by means of convection and radiation methods. The results have proved a direct dependence between the surface temperature of the radiator and the floor heating area and the intensity of the induced heat flow due to the temperature profile. By increasing the radiator temperature, we can increase the output heating power delivered to the external space, but concurrently we can increase the air flow intensity. As the optimal ratio of the floor heating surface output power to the radiator output power for the above-described mathematical model, we have obtained a value of 70%/30%. REFERENCES Petráš, D.: Floor Hot-water Heating, Jaga Group, Bratislava 1998 Petráš, D., Koudelková, D.: Hot-water and Electric Floor Heating. Jaga Group, Bratislava 2004 Berounský, V.: Radiators in Theory and Practice. (Heating Industry 16.) Prague, ČSVTS 1987 Halahyja, M., at al.: Thermal Engineering Techniques, Acoustic and Lighting Systems in Buildings. Bratislava, ALFA - SNTL 1985 ANSYS, Inc.: Reference Manual 19 mikula_01.indd :15:50