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Application of the Finite Element Method for Modelling of District Heating Network Irina GABRIELAITIENE* PhD student Lund Institute of Technology Lund, Sweden

Graduate of Vilnius Gediminas Technical University, Engineer’s Degree in Heating (1994), Master’s Degree in Information Sciences (1996), PhD student (since 1997) Dept. of Strength of Materials Saulėtekio al. 11, 2040 [email protected]

Rimantas KACIANAUSKAS Professor Vilnius Gediminas Technical University, Vilnius, Lithuania

Graduate of Vilnius Gediminas Technical University (1975), PhD (1982), Professor (1997) Author of 1 monograph and over 80 papers. Research interests: application of the finite element method in civil, mechanical and thermal engineering. Dept. of Strength of Materials Saulėtekio al. 11, 2040 [email protected]

Bengt SUNDEN Professor Lund Institute of Technology Lund, Sweden

Graduated as a M.SC (1973) and presented PhD (1979) at Chalmers University of Technology. Author of 175 papers and 2 textbooks. Dept. of Heat and Power Engineering Box 118, SE 221 00 [email protected]

Construction Informatics Digital Library http://itc.scix.net/ paper ecce-2001-9.content

*permanent address

Summary This paper presents simultaneous modelling of the fluid flow and heat transfer in a district heating network. The finite element method is applied here. A major advantage of the finite element method is the efficiency of verified numerical procedures and standardized information describing the fluid, which may be transferred and communicated with other computer aided design or monitoring systems. The thermal-hydraulic pipe element of the ANSYS code is modified for these purposes by modelling the real structure of the pipeline, the heat conductivity and convection. The finite element model is compared to the traditional H.Cross method. A fragment of the district heating system of the city Vilnius, Lithuania is used as an illustrative example. Keywords: flow and heat transfer, finite element method, district heating network.

1.

Introduction

The district heating (DH) pipelines, often transporting heat over long distances and distributing it to large built-up areas, unavoidably give rise to losses in both heat and pressure. The pressure losses require the addition of pumping power to the DH system. Heat losses are also undesired phenomena which increase the primary energy consumption for a given size of the heat supply to the buildings connected to the system. Since DH pipelines are very costly, it is very important to develop an 1001

efficient modelling method by which the system behavior may be determined and which has the ability to foresee consequences of various decisions in design. Therefore, the present study is focused on more advanced numerical modelling of simultaneous fluid and heat flow. The fluid flow in a pipe network poses a problem of a nonlinear nature. This is due to the fact that the dependence of the flow rate and the pressure is nonlinear. The main difference between methods of solving this problem lies in the selection of independent variables. The oldest and perhaps the most widely used method for analyzing pipe networks is the Hardly Cross method [1]. The Hardly Cross method deals with each loop independently providing no direct interaction between the basic network equations. This is the major reason for the H. Cross method being frequently found to converge too slowly, if at all [2]. Zhao [2] investigated the concerned properties of three methods for steady-state flow analysis of DH networks: general Newton-Raphson method, the linear theory method and the basic circuit method. Almkvist [3] adopted the method of characteristics to analyse complex pipe systems and simulated steady as well as transient flow behavior. Valdimarsson [4] implemented the graph-theoretical method by the Matlab code, called Pipelab. The graph-theoretical model solves the non-linear equations for the loops, and flow solution of the network flow is obtained by the linearisation method. According to reference [5], two methods, the element method and the node method, have mainly been used for physical modelling and simulation of temperature dynamics and fluid flow in district heating networks. Wallenten [6] presented formulae for district heating pipes by using the multipole method, which can solve two-dimensional steady-state heat conduction problems with circular boundaries. Schneider [7] proposed formulae for the heat loss from a pipe with insulation and a ground surface thermal resistance, based on computer computation with a finite difference method. Bohm [8] studied the dynamics of a system with more than a single pipe. The above mentioned thermal-hydraulic problems have no analytical solution but have to be solved numerically. Different approaches and numerical techniques may be used but the finite element method seems to be the most universal and powerful tool used in engineering. The finite element code ANSYS [9,10] is used here for simultaneous modelling of the fluid flow and heat transfer in a district heating network. This code suggests not only required elements and well tested solution methods but also provides efficient pre- and postprocessing of data, which is very important for multistage computer-aided design as well as control of the district heating network.

2.

Numerical approach

2.1 Thermal-Hydraulic Model Generally, the thermal-hydraulic model used for description of the simultaneous fluid and heat flow in piping network presents a coupled non-linear problem. This implicit model requires simultaneous solution for the temperature field because the pressure conductivity matrix contains temperaturedependent fluid physical properties, such as fluid mass density and fluid viscosity. The thermal analysis represents the heat balance and reflects thermal conductivity, convection and mass transport, where the thermal conductivity matrix (thermal conductivity and film heat transfer coefficient) depends on flow rate and temperature difference. Finally, the coupled thermal-hydraulic finite element model may be written in the following form: (1)

0   P  W   K(P, T)  =   0 K(W, T) T   Q  

where: {T }- nodal temperature vector; {P} - nodal pressure vector; {Q} - nodal heat flow vector; {W } nodal fluid flow vector [K(W,T)] - thermal conductivity matrix (includes effects of convection and mass transport); [K(P,T)] - pressure conductivity matrix. In an uncoupled thermal-hydraulic model mechanical and thermal properties of fluid do not depend on each other. Pipe flow problems are non-linear because the flow conductivity matrix depends on the pressure difference. The convection and mass transport, determined by the thermal conductivity 1002

matrix, is related to the fluid flow rate. The flow rate and temperature gradient are solved through an iterative process in which the conductivity matrices are updated in each iteration to reflect the new pressure difference. For an uncoupled thermal-hydraulic model the equilibrium equations have the form of: (2)

0   P  W   K(P)  =   0 K(W) T   Q  

d

2.2 Elements Formulation and solution of the uncoupled model, written in equation (2), requires computation of the conductivity for the entire piping network by using finite element (FE) technology, this procedure involves computation of individual elements and standard assembling procedure. The main problem is that the existing FE library of the ANSYS code does not contain element capable to describe the fluid and heat flow in insulated pipe. Therefore, a novel combined approach for the problem (described in equation (2)) formulation is required while the standard procedure is used in the solution stage. The ANSYS library contains pipe element for simultaneous modelling of flow in the pipe, where thermal convection through the fluid boundary is described by a film heat transfer coefficient. Additionally, different thermal elements are suggested for the user. The novel complex thermal-hydraulic pipe element for Tp simultaneous modelling of uncoupled fluid and heat Convection flow in insulated pipe is suggested in the present Q2 element TN report. The element presents assembly of standard N z elements describing not only flow but also heat Conduction element transfer through the insulation into the environment. It TL TO contains standard pipe element where a series of oneL x dimensional thermal elements are connected to the Q1 pipe nodes. The complex conduction through the J TM insulated multi-layered structure is determined by a M conduction element (Fig.1), where thermal conduction Q of an individual layer is defined by the thermal TK K coefficient Ki. The convection element is used to Fluid element determine convection and radiation effects from the surface of the insulated structure by a convection I TI coefficient hi, which depends on air temperature and L W temperature on the external surface of insulation. Fig. 1 Uncoupled finite element model The existing software capability is modified to simulate the behaviour of fluid flow in an insulated pipe by the uncoupled model, as it is formulated in real engineering. In general, the software has an existing set of finite elements, which allows simulation of only separate processes, such as heat conduction, heat convection and fluid flow. Thus, the main task was to found the suitable elements and put these elements in a certain model, in such way that they could reflect processes, occurring in a district heating network. At first, the fluid element to simulate fluid flow was found. Besides that this element provides flow analysis, it can simulate the following processes: (i) the conduction and convection occurring in the fluid and in the fluid layer adjacent to a surface, (ii) heat flow due to conduction within the fluid and the mass transport of the fluid in the pipe longitudinal direction. Hence, this element cannot simulate behaviour of the fluid with adjacent insulation and heat exchange through this layer to the surrounding air. In connection to this, the three elements (fluid element, conduction and convection element) were coupled so that they could work together and exchange variables with each other. After solution of equation (2), computation of the heat loss through the insulated structure (Q1 and Q2) (Fig.1) by using already computed an element temperature was a secondary problem. Air

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temperatures (To and Tp) (Fig.1) and initial temperature of the liquid (TI) are assumed known. In this study, heat is transferred to air in a duct; but heat transfer through the duct and to the soil is not considered. 2.3 Case study A case study is performed to demonstrate the idea discussed in the previous section. In the present study, data from the Ø530 Ø325 district heating system in Vilnius, (330m) Ø325 (200m) Ø530 (130m) Lithuania is used. Figure 2 shows (280m) the distribution network of the Ø530 Ø325 (100m) Ø530 system. There are one DH plant (620m) (468m) and 27 heat exchanger substations Ø159 Ø530 Ø530 Ø530 (240m) (40m) (235m) (209m) (consumers). The DH network is Ø720 Ø530 (155m) set up of insulated pipes (mineral (109m) Ø530 wool), which are covered in a (177m) Ø530(89m) common concrete duct. The pipe Ø530 Ø720 Ø219 (297m) dimensions range from 159 mm to (500m) (100m) 820 mm and all connected Ø219 (64m) Ø273 buildings are situated within an (18m) area of 3.6 km2. Ø530 Ø273 Ø720 (255m) (301m) (92m) Ø219 Steady state analysis of the DH (196m) Ø530 (223m) network is referred to in the following as finding the flow and Ø530 (409m) Ø530 (344m) Ø273 heat losses distribution in a Ø720 (280m) Ø720 (70m) (313m) network when the network Ø159 (134m) structure, the flow demand at each Ø720 (177m) Ø530(560m) consumer are known. In this Ø325 (170m) Ø426 study, we consider the supply (49m) pipes. Heat is transferred through Ø159 Ø720 (284m) (32m) Ø530 water at initial temperature T at the district heating station and (369m) Ø720 Ø219 having temperature the air in the (70m) Heat Exchanger ducts. The fluid physical Ø530 (214m) Ø325 Ø159 Station Ø530 (252m) (260m) properties, such as fluid mass (540m) Ø720 (169m) Ø530 District Heating (136m) density, specific heat, and fluid Station Ø273 viscosity are known. The non(154m) Ø820(720m) Ø530 linear material properties table is (231m) Ø530 Ø325 (305m) used to provide the flow(102m) dependent film heat transfer coefficients. Fig.2. Vilnius district heating system 2.4 Finite Element Model Numerical discretization of the district heating system is a complicated and time-consuming task. The problem is that presentation of a pipe section in terms of finite elements is not unique and a strongly mathematical procedure. Sometimes knowledge-based sophisticated arguments are of the major importance. Discretisation of the pipeline means both subdivision of the pipe section in the longitudinal direction by one-dimensional pipe elements as well as subdivision of the insulation in the radial direction by one-dimensional thermal elements. The influence of discretization as well as numerical error by a series of numerical experiments was investigated. The results of the numerical experiments are used in the development and evaluation of the thermal-hydraulic finite element model and may be summarised as follows: (i) The relative accuracy of heat flow decreases when the insulation thickness decreases, but remains independent of thermal conductivity of the insulation material. Acceptable number of thermal elements through the insulated structure is defined and recommendations for practicing engineers are suggested. (ii)Recommendations for longitudinal subdivision of the pipe section into pipe elements are

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suggested. It is not necessary to subdivide the pipe section with a length up to 1 km. On the basis of the above recommendations the user-friendly data pre-processing technology is developed. Numerical solution of the thermal-hydraulic model (described in equation (2)) followed by corresponding post-processing of the results, reflects the flow behaviour in the network. The most important fluid characteristics are the pressures drop in each node of the network and the flow rate for individual pipes. The thermal behaviour is defined by the temperature distribution of water, temperature distribution in insulation layer and temperature on its surface, as well as the heat losses at each node of the network. Besides, the heat conduction components and transmission fluid in the pipe longitudinal direction have been defined. Additional information, such as Reynolds, Prandtl, Nusselt numbers and friction factors in each pipe are determined. The output of the calculation model can be postprocessed and represented graphically in a number of ways, showing pressure, heat losses in each node, flow in each pipe of the network, as well as other additional parameters.

3.

Results and discussion

Temperature

Results for the thermal problem, which is the major target in this study, are presented in graphical forms. T=const Figs. 3 and 4 show the comparison between the 1.00 0.990 current results and results by Tuomas et al. [1]. 0.988 0.98 The temperature distribution, in Fig.3, and heat flow, in Fig.4, in the network, for the different 0.96 lengths (1km and 10km) respectively are present in graphical forms. The numerical experiment is 0.94 carried out with and without the assumption that Analytical,L=1km the fluid temperature in the pipe network is 0.92 Analytical,L=10km 0.913 constant. FEM,L=1km 0.90 FEM,L=10km The influence of simultaneous thermal-hydraulic 0.894 solution on the flow behaviour shows a decrease 0.88 0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000 of the temperature level of the district heating Length network (Fig. 3). Fig. 3. Comparison of temperature distributions between (FEM) and analytical method

Heat flow rate

The comparison between numerical simulation and analytical solution shows a decrease of the value of the heat losses in all distribution networks (Fig.4). 1 This thermal-hydraulic solution has influence 1.0 0.936 Analytical 0.931 on the results of district heating solution, because the calculation of the heat losses in a FEM, (T=const) 0.885 0.8 node takes into account the temperature of the FEM,L=1km FEM,L=10km fluid in this node. The temperature of the second node depends on heat losses from the 0.6 previous node. Thus, this sequence provides an accurate determination of the heat losses of district heating network. In cases, where district 0.4 heating pipelines transport heat over long distances and distribute it, heat losses decrease. 0.2 (Fig.4). The obtained results reflect the 0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000 influence of the temperature drop in the Length network neglected by calculations using traditional engineering approach. Fig. 4. Comparison of heat flow between finite element (FEM) and analytical method 3.1 Conclusions The following conclusions were drawn from this study:

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• • • • •

The finite element method was shown to be a powerful engineering tool for the modelling of district heating systems. Strong mathematical formalism, verified efficient solution algorithms and standard computer-aided technology may be applied for simulation of fluid and heat flow behaviour. The compared solutions of the current results and results by Tuomas [1], for practice show decrease of heat losses and temperature distribution in network, because of the influence of simultaneous thermal-hydraulic solution. The proposed complex thermal-hydraulic pipe element is supported by practical recommendations and data preprocesssing technology, which may be used for practicing engineers. The method is independent of network geometry and standard discretisation methodology includes the behaviour of the entire network, but not only individual branches or nodes. The proposed finite element model presents a separate tool which may be used at different stages of design and to monitoring of a district heating network. The proposed pipe element, discretisation technology and experience gained, may be used not only for simultaneous solution of uncoupled fluid and heat flow but can be extended to different kinds of coupling and non-linearities of steady-state and time-dependent flow.

3.2 Acknowledgements One of the authors (I.Gabrielaitiene) thanks The Swedish Institute for providing a scholarship. The authors (I.Gabrielaitiene, R.Kacianauskas) would like to thank Vilnius District Heating Company (Vilniaus Silumos Tinklai) for providing the data that being used in the study. 3.3 [1] [2]

References Tuomas E., Gedgaudas M., Heating Supply, Vilnius (in Lithuanian), 1993, 328pp. Zhao H., Analysis, Modelling and Operational Optimisation of District Heating Systems, Centre for District Heating Technology, Laboratory of Heating and Air Conditioning, Technical University of Denmark, Lyngby. 1995, 226pp. [3] Almkvist S. Calculation of Transient Flow in Complex Pipe Systems. Thesis for the degree of Licentiate of Engineering. Chalmers University of Technology, Department of Thermo- and Fluid Dynamics, Gothenburg, 1991, 63pp. [4] Valdimarsson P., Korsman J.”Use of a Graph-Theoretical Calculation Model For Water Flow in District Heating Systems – Experience form Nuon, Holland”. Proceedings of the 6th International Symposium on District Heating and Cooling Simulation, Reykjavik, Iceland, August 28-30,1997, pp.1-15 [5] Palsson H. “Analysis of Numerical Methods for Simulating Temperature Dynamics in District Heating Pipes” Proceedings of the 6th International Symposium on District Heating and Cooling Simulation, Reykjavik, Iceland, August 28-30,1997, pp.1-19 [6] Wallenten P., Steady-State Heat Loss from Insulated Pipes, Dep. of Building Physics, Lund Institute of Technology, Sweden, Lund, 1991, 198pp. [7] Schneider G.E, “An investigation into heat loss characteristics of buried pipes”, Journal of Heat Transfer, Transactions of ASME, vol 107, August 1985, p696-699. [8] Bohm B, Energy-Economy of Danish District Heating Systems, Laboratory of Heating and Air Conditioning. Technical University of Denmark, Lyngby, 1988, 229pp. [9] ANSYS. Theory Reference. Release 5.4. 1997 SAS IP, Inc.. [10] ANSYS. Elements Reference. Release 5.4. 1997 SAS IP, Inc..

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