12th International Symposium on District Heating and Cooling

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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
The <strong>12th</strong> <strong>International</strong> <strong>Symposium</strong> on District Heating and Cooling,September 5 th to September 7 th , 2010, Tallinn, EstoniaEVALUATION OF MEASUREMENTSThe temperature-dependent thermal conductivity of thepolyurethane foam is obtained by calculating thetemperature decline of the coil water. Certain values ofthe thermal conductivity of the polyurethane arechosen, λ 50 and λ‟. The actual λ(T) are obtained byminimizing the difference between the measured andcalculated coil temperatures, (4).The difference between the calculated and measuredtemperatures for the optimal parameter values of λ 50 , λ‟and c are showed in Figure 9. The saw toothdisturbance from the measurements is seen. Thedifference giving the best fit lies in the interval -0.20 to0.25 (ºC). The error is small.The heat capacity c (J·kg -1·K -1 ) of the polyurethanefoam is input to the calculations. Literature referencesfor the heat capacity of polyurethane foam varies, 1300J·kg -1·K -1 at 50ºC in [9], 1400 J·kg -1·K -1 in [10], 1400-1500 J·kg -1·K -1 for rigid polyurethane foam in [11].The densities and heat capacities of water, service pipeand insulation were assumed to be constant in thetemperature interval studied.The optimal parameter values of λ 50 λ‟ and c wereobtained by minimizing the difference D (ºC) betweenthe calculated T w (ºC) and measured coil temperaturesT w,meas (ºC).D max T ( t) T ( t)for t t t (4)w, calc w, meas1 2A certain time interval, 0.5
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academic access is facilitated as t
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1. CHP system operation in A2. Ther
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is covered by operating HOB. In oth
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produce heat and electricity. Fluct
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To assure that the temperatures mea
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production and provide for marginal
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In addition, it can also be observe
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owner is normally only interested i
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Stockholm district heating system a
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The values presented do of course l
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12th International Symposium on District Heating and Cooling

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