12th International Symposium on District Heating and Cooling

The <strong>12th</strong> <strong>International</strong> <strong>Symposium</strong> on District Heating and Cooling,September 5 th to September 7 th , 2010, Tallinn, EstoniaMODELLING USING CONFORMAL COORDINATESIt is rather complicated to calculate the temperaturedecline in twin pipes due to the pipe geometry. A socalled conformal mapping presented in [8] was used tomap the twin pipe geometry onto a rectangulargeometry. In the experimental measurements, thesupply and return service pipes were assumed to haveequal temperatures in the test-procedure. Then,symmetry is assumed between the four quarters of apipe cross-section. A quarter of a twin pipe is studied,see Fig. 5. In the x,y-plane, the temperaturedevelopment is described by the heat equation:T T T c ( ( T) ) ( ( T) )t x x y y(1)The (x,y)-coordinates ( z x i y)are transformed tosuitable conformal coordinates ( w u i v)with theaid of line sources and so called multipoles.water and the right-hand boundary against the pollwater. The heat flux in the vertical v-direction is zero onthe horizontal boundaries due to symmetry.Fig. 6 Initial temperature distribution in the crosssectionof a pipe quarter in the u, v-plane.In the numerical solution, the region is divided into arectangular mesh. The area factor is now the area ofeach of the cells shown in Fig.5. They are shown inFig. 7. The largest cell is the one in the lower left cornerin Fig.5 near the stagnation point (usp). The areas areused to calculate the heat capacity of each cell in the u-v-plane.Fig. 7 Areas of the computational cells in the x, y –planetransferred to a u, v-plane. The stagnation point is denotedusp.Fig. 5 A quarter of a twin pipe in x-y-plane geometryThe heat equation in the conformal coordinates is:T T T c A( u, v) ( ( T) ) ( ( T) )t u u v vHere, A(u,v) is the area factor in the conformaltransformation.The considered region shown in Fig. 5 is transformedto a rectangular region in the u, v-plane, see Fig. 6. Inthe figure, the left-hand boundary lies against the coil(2)The initial steady-state condition for a twin pipe withcoil water temperature T w = 81.3ºC immersed into poolwater at T0=19.7ºC is showed in u-v coordinates inFigure 6. Then, the temperature decline of stagnantwater in the twin pipes are calculatedThe density ρ and the heat capacity c of thepolyurethane foam are assumed constant in thetemperature interval studied. The boundarytemperatures at the casing are given by the pooltemperature.The thermal conductivity λ(T) of the polyurethane foamis determined by the thermal conductivity at 50ºC λ 50(W/m·K) and a coefficient λ‟ to account for a lineartemperature dependence.50 50( T) 1 ' ( T T )(3)93