© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in , B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

Analysis of fluid flow andheat transfer in district heating pipelines using the finite element method

I. Gabrielaitiene1'2,B. Sundin' & R. Kacianauskas2 'DepartmentHeat and Power Engineering,Division of Heat Transfer, Lund Institute of Technology,Lund, Sweden. 2VilniusGediminas Technical University,Lithuania

Abstract

The finite element method is applied to solve coupled steady-state fluid flow and heat transfer problems in insulated pipes of district heating networks. The main advantage of the finite element method involved here is the standard discretization and solution technique for both coupling of temperature-dependent physical properties in the thermo-hydraulic flow andfor description of the multilayered structure of insulation, which may also be situated in the environment, such as soil or air. In order to describe the above problem in a standard manner, the complex thermo-hydraulic macroelement has been proposed. This element presents a combination of a standard hydraulic pipe element and one-dimensional thermal elements of the ANSYS code. The proposed approach isused to investigate temperature-dependent fluid properties and the role of the coupling in district heating pipelines.

1 Introduction

Prediction of pressure and heat losses is an important consideration in design of district heating (DH) pipelines, because the energy transfer will affect the environment in the surroundings of the buried pipe. Pressure losses require the addition of pumping power to the DH system. Moreover, the energy transfer to or from the pipe will affect pumping requirements of fluid flow in a pipe. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

Numerical methods applied for design of district heating pipelines in the Nordic countries are based on engineering simplifications [1-2]. Such simplified approaches contain difficulties due to the multilayered, even multisolid, structure of the pipeline and the environment. Moreover, the fluid physical properties, involved in both fluid flow and heat transfer analysis, depend on temperature and this aspect is important in design calculations. Thus, fluid flow and heat transfer analyses are coupled due to temperature variation of these properties. Limited effects of coupling are implemented into a popular code TERMIS which is used in engineering practice [3].Despite large amount of publications in heat transfer analysis [4] applications of the finite element method in heating systems are still limited. A finite element formulation for the solution of hydraulic problem in piping systems was presented in [5]. The heat flow analysis in a pipe considered as an axisymmetric insulated cylinder is illustrated in [6]. Usually, the of the water-pipe interface and resistance of the pipe wall are disregarded [7]. The problem of heat loss from buried pipes including convection coefficients on both the interior surface and on the exposed ground above the pipe is presented in [S]. The emphasis of this report is placed to develop a general finite element approach to model coupled temperature and flow fields in insulated pipelines of district heating networks. Another objective is to examine the above problems, considering the selected temperature dependent fluid physical properties and thermal resistance of a multilayered structure of the pipelines.

2 Description of the district heating pipelines

Usually, a district-heating system is presented as a grid composed of one- dimensional segments. Important external parameters such as fluid flow rate can be set either to nodes of the grid or to its segments. The load or demand of heat energy is generally expressed in terms of heat flow at the nodes. The most common pipes in use for district heating are insulated pipes with carrier of steel, insulation of polyurethane and a casing of high-density polyethylene (Figure 1).

Figure 1: Common insulated pipe.

Usually, the heat transfer from the fluid to environment is a multistage process directed from a warmer fluid to a wall, then through the multilayered wall to a colder environment, etc. The hydraulic problem of the fluid flow is mainly controlled by the friction coefficient, representing the friction between flowing fluid andthe pipe. It also includes a separate loss coefficient, which © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advancc.d CompututiodMcthods in HatTrumfc.r 15 represents fittings (i.e., elbow) in DH pipelines. The difference between district heating water temperature level and ambient temperature of the environment is the driving force for the heat transfer. In water-based district heating systems three parameters, i.e., the flow and return temperatures as well as the fluid flow rate describe the quantity and quality of the heat provided. The water temperature inuse for district heating systems is varying from 25°C to 120°C. In this interval physical properties ofwater are varying but differently. As shown in Figure 2, density of water is decreasing by 5.8%, is increasing by 1 ].l%,while specific heat is increasing by 1.7% [S]. A strong temperature dependence of the viscosity gives a decrease by 3.7 times in the above temperature interval. Generally, the solution of thermo-hydraulic problems is presented in terms of pressure losses and temperature fields, but heat losses, velocities and other parameters important in engineering have to be obtained by accompanied computations. Lkwl LWm wsccslly LW

97~ Y72~ %7cii x% 0.472 957 - 0.412 952 - 0.352 “7 0.292 0.232 -l 25 45 65 85 105 T 12.5 25 45 65 85 105 ~125 Wkrn WCM

4.241 ~

4.233~

4.224 ~

4.216~

4.m~ 4.19-

4,M~

25 45 65 85 K6 ~125 Figure 2: Variation of some properties of water with temperature.

3 Mathematical model

3.1 Coupled approach

Generally, the fluid flow in a pipe and heat transfer involving conduction and convection is a simultaneous phenomenon. The finite element approach to this coupled steady-state non-linear problem gives a mathematical model in the form of equilibrium equations, which may be written in the following form: © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

Here, {P}and { T} present nodal pressure and temperature vectors, which have to be obtained by solution of the problem (1). Nodal fluid {W} and heat flow vectors {Q}describe given external loads and boundary conditions, while [Kh] and [KJ present hydraulic and thermal matrices reflecting physical properties of the fluid and the insulation. The first equation describes non-linear hydraulic flow, where the hydraulic matrix [Kh]depends on the values of pressure difference. The second equation presents the thermal problem, where the thermal matrix [Kt] reflects conductivity, pressure dependent mass transport and convection. Fluid physical properties, such as density, viscosity, specific heat and thermal conductivity are temperature-dependent and involved in both heat transfer and fluid flow analysis. Hence, fluid flow and heat transfer are coupled due to variation of these properties and the coupling is built into the governing equations implicitly. The solution of model (1) requires an iterative sequential procedure. The pressure and temperature gradients are solved through an iterative process in which the matrices are updated by iterations.

3.2 Uncoupledapproach

The uncoupled thermo-hydraulic approach considers mechanical and thermal properties of the fluid independently. It is assumed that the temperature gradient does not affect velocity and temperature profiles. This is equivalent to specifying that the properties of the water (density, viscosity, specific heat, thermal coefficient) remain constant in the flow. For an uncoupled thermal-hydraulic model the equilibrium equations have the form:

The fluid flow problem (2a) remains non-linear, while the thermal problem (2b) becomes linear due to assumption of constant water properties. After solution of eqn (2a), the obtained pressure vector is used as an input in eqn (2b).

4 Finite element model

4.1 Thermo-hydraulicmacroelement

The main difficulty in a thermo-hydraulic finite element analysis of the pipe is thatthe existing finite element codes usually contain thermo-hydraulic pipe elements paying no attention to the pipe structure, which contains different material properties. By neglecting that structure, it is impossible to evaluate thermal properties and actual boundary conditions under which heat transfer from the pipe occurs. The segment of the pipe is considered here as a thermo-hydraulic pipe macroelement designed in terms of a multilevel modelling approach [lo]. The developed complex thermo-hydraulic macroelement contains fluid element and © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advancc.d CompututiodMcthods in HatTrumfc.r 17 thermal elements such as conduction and environment. Geometry of the complex macro element is defined by nodes l and J, while for surrounding nodes K and L are used. Inlet temperature is described by TI , while outlet temperature by TJ (Figure 3). The temperatures of environment TKand TLare assumed to be known.

4.2 Element matrices

Eqn (1) aswell as eqns (2a) and (2b) present the model of the entire piping system, where matrices [&l and [K,] are formed by assembling the individual elements. For an individual element, in this case macroelement e, the hydraulic matrix mainly depends on the fluid properties, including fluid-pipe interaction. This matrix expresses frictional pressure losses along the element [Kt] and a separate loss at nodes of element [K,].Symbolically this is represented as:

Estimation of the frictional pressure losses is provided in [ 1 l], where the friction coefficient includes temperature-dependent viscosity and density. For the uncoupled approach, the above mention properties remain constant.

Environment T,

Figure 3: Complex thermo-hydraulic macroelement.

The physical nature of the thermal matrix is much more complicated. The thermal matrix reflects heat transfer through a multilayered, or even multisolid, structure of the pipe and environment. For an individual element e it may be represented as a sum of different layers:

LK;l=~lK:l. (4) m=i In eqn (4) subscript m denotes all layers of insulation and environment producing thermal resistances. On the other hand, individual layers may contain different transfer modes. Each layer may produce a thermal matrix:

Here, the [Kcnv(T)]temperaturedependent matrix describes convection to the layer adjacent to a surface, which mainly depends on the convection film © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

coefficient h(T), [K7,,,s,(P)] reflects mass transport, while [K,,dl,(T)] and [Kc,,&')] are temperature-dependent matrices representing conduction in the flow longitudinal direction and normal to the flow direction, respectively. The fluid element is a thermo-hydraulic element, which describes fluid flow and heat transfer from a fluid to a wall. The behaviour of the flowing fluid is described by the hydraulic matrix [K;,] in eqn (3). The thermal component is described by the thermal matrix [K,]and is governed by temperatures: energy- average fluid temperature across the pipe and temperature of the layer adjacent to the internal surface of the pipe at the nodes of fluid element. The fluid represents a single layer m and the thermal matrix is given as a particular case of eqn (5):

[Ki,l=[K,n,(T)]+[K,,,,(P)I+[Kcnd~(T)l. (6) The convection component [&,,(T)] is estimated through convection film coefficient h(T),which may be evaluated according to [12]. Viscosity, thermal conductivity and specific heat as functions of temperature are included in the estimation of the convection film coefficient. Estimation of the conduction component [Kcnd~(T)]incorporates temperature-dependent thermal conductivity. Moreover, a set of conduction elements is required to represent the transfer of heat in the pipe, insulation and casing material. Because properties of each material are different, thermal matrices of different properties describe the thermal resistance of each layer. For describing conduction in the insulation, the following matrix is used: [Kt~l=[&ndR 1. (7) For the environment layer, the thermal matrix depends on the nature of the layer. In a case of soil, conduction is only taken into account and the thermal matrix (7) remains. The soil is considered as a single layer and depth of the buried pipe is taken as the thickness of layer. For description of air, combined effects of thermal radiation and natural convection from the insulation surface are taken into account. Then the thermal matrix has the form: LK7J=Kw1' (8) Evaluation of thermal properties of the environment is a specific task and may be found in the references of [1l]. Evaluation of heat losses has to be performed within individual elements e:

m=1 Finally, the heat loss of the network is found from the sum of losses within individual elements e.

4.3 Composition of element

Subdivision of insulation layers into thermal elements as well as subdivision of pipe segments into pipe elements require additional investigations, which have been done by numerical experiments. The number of elements depends on the insulation properties and have to be estimated by © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advancc.dCompututiodMcthods in HatTrumfc.r 19 evaluation of the heat loss. Longitudinal subdivision of the pipe does not influence the results of the hydraulic solution (i.e., pressure losses), while it may significantly affect the results of thermal analysis. The thermo-hydraulic macroelement is created by assembling of conventional elements of the ANSYS code [ 131.

5 Comparison between coupledand uncoupled model

Numerical examples have been considered for both coupled and uncoupled models and comparisons have been made. The influence of coupling by assuming a temperature dependent variation of density and viscosity, thermal conductivity and specific heat has also been investigated. The properties of water are chosen at an average water temperature of To= 80 "C. Numerical experiments have been performed for a pipeline with diameter of the pipe D = 0.3 m and various temperatures at the inlet of the pipe TI.Properties of water are considered from Figure 1 in the temperature range from 40 "C to 120"C. The heat conductivity of the steel pipe is 76 W/mK, while the heat conductivity of insulation and casing is 0.04 and 0.43 W/mK, respectively.

5.1 Influence of temperature of incoming water

The outlet temperature TJ and pressure losses have been investigated for a pipeline having the length L = 20 kmwith different values of the inlet temperature TI,such as 60 80, 100, 120 "C. The relative differences between the coupled and uncoupled approaches are presented in Figure 4, where the zero line represents the uncoupled approach. As is shown in Figure 4, in a case of positive difference between the inlet and reference temperatures, the temperature at the outlet of the pipe increases due to the influence of the coupling, while the frictional pressure loss decreases.

4 b) Frictional pressure Temperature in the outlet losses of the pipe %

oycoupled

Coupled O.Oj* 0.03 /o -0.0 12!-4 60 80 100120 120100 80 60 Temperature of incoming water Temperature of incoming water

Figure 4: The influence of different temperatures of incoming water on the results: a) outlet temperature TI;b) pressure losses P. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 20 Advmced CompututiodMethods it1 HwtTrmsfkr

5.2 Influenceof length of the pipeline and temperature of incoming water

Influences of the pipe length have been investigated by considering different lengths L = l h,10km and 20 km and the inlet temperature TI varying from 100°C to 60°C. As can be seen from Figure 5, as the length of the pipeline is increased, the temperature at the outlet of the pipe is increased but the inlet temperature TIdoes not play an important role. Vice versa, for the results of the frictional pressure losses, the temperature of the incoming water as well as the length of the pipe play important roles. As the temperature at the inlet of the pipe is higher than the reference temperature, the difference between the results increases for larger length of the pipeline.

T=lOO'C

T 0.0 S. 0'024%Uncoupled - ?? . 0.003% e rn-l I -?l.O02% -0.012% a $ : I -0.0 m + - 0 T=600C

0 1 lo Length [km] 0 10 Length [km] 20 Figure 5: Variation of difference between coupled and uncoupled approach with length and temperature.

5.3 Influenceof the water physical properties separately

Solutions for the case of a district heating pipeline length L = 20 hhave been carried out with inlet temperature T, = 100°C. The density p, viscosity p, specific heat c and thermal conductivity K have been considered as functions of temperature separately and the influence of each of them on the results is presented in table 1.

Table 1: The influence of the water properties on the frictional pressure ksses, P and temperature at the outlet of the pipeline, T. YV) P67 PVL CV) KU3 P(T),PV'), PG7 cm>Km Press.,P -4.340% 0.715% -3.623% -0.001% -0.001% -3.6~5% Temp., T -0.001% 0.002% -0.001% 0.039% 0.001% 0.040%

Simultaneous coupling of the density p and viscosity p causes variation of the pressure losses up to 3.6%, while the variation of the outlet temperatures 0.039%is caused by the heat capacity c of water. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2 Advancc.d Compututiod Mcthods in HatTrumfc.r 2 1

5.4 Case of study

A fragment of the district heating system of the city of Vilnius (Lithuania) is used as an illustrative example to demonstrate the ideas discussed in the previous sections. Steady-state analysis of the DH network is performed. The results of simulation of a real network by the uncoupled approach and a comparison with analytical solution have been presented in [ 141. The simulation results for a real network by the coupled approach show insignificant differences in frictional losses and temperature with respect to a single segment.

6 Discussion and conclusions

The presented finite element approach and proposed thermo-hydraulic macroelement is a suitable tool for the investigation of temperature-dependent coupling in district heating pipelines. On the basis of numerical experiments performed for the segment of the pipelines, the quantitative influence of coupling on the fluid flow and heat losses is estimated and the following conclusions are drawn: l For the length L = 20 kmof the pipeline, the gives an insignificant difference of 0.12% in the outlet temperature, while the difference in the pressure losses reaches more than 9% and has to be taken into account. l The difference between coupled and uncoupled models for the length of the same pipe ranging form L = 1 km to L = 20 km and for the inlet temperature varying between 60 "C and 100"C remains insignificant for the outlet temperature. The difference between the models in the pressure losses depends on the difference between inlet and reference temperatures. For a negative difference, when TI = 60 "C, the pressure losses are increased, while for a positive difference, when T,= 100 "C, the pressure losses are decreased. l The variation of pressure losses for the segment of the pipeline is caused by simultaneous coupling of density and viscosity of water, while the variation of outlet temperature is caused by the specific heat of water.

Acknowledgement

The first author kindly acknowledges the financial support from Nordic Energy Research Programme.

Nomenclature

{P}- nodal pressure vector ,U - viscosity [kg/ms] { 2') - nodal temperature vector L - pipe length ~~1 {W} - nodal fluid vector D - internal diameter if pipe [m] {Q}- heat flow vector A - cross sectional area of the pipe [m'] [Kh]- hydraulic matrix f- friction factor (-) [Kt]- thermal matrix Z - single loss coefficient (-) K - thermal conductivity [W/mK] © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advanced Computational Methods in Heat Transfer, B Sunden and CA Brebbia (Editors). ISBN 1-85312-906-2

c - specific heat [J/(kgK)] TI- temperature in the inlet of the pipe ["C] h - film coefficient [W/m2K1 TJ- temperature in the outlet of the pipe["C] p - density [kg/m3] To- reference temperature ["C]

References

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