I. Gabrielaitiene* and R. Kačianauskas Department of Strength of Materials
Vilnius Gediminas Technical University, Vilnius, Lithuania e−mail: irena.Gabrielaitiene@fm.vtu.lt
B. Sunden
Department of Heat and Power Engineering Lund Technical University, Lund, Sweden
e−mail: Bengt.Sunden@vok.lth.se
ABSTRACT
Summary The steady state thermo-hydraulic flow in a pipe is investigated by the finite element method, where the main attention is focused on analysis of heat losses in a multilayered structure of a pipeline. A complex thermal finite element has been developed for this purpose and combined with thermo-hydraulic pipe element of the ANSYS code. The results for heat losses are presented and compared to analytical solution. Accuracy of the finite element model is also investigated.
Introduction
Evaluation of heat losses presents an important part of thermo-hydraulic flow analysis in engineering applications such as district heating and cooling systems. The solution of this problem contains difficulties due to the multilayered, even multisolid, structure of the pipeline and due to different heat transfer modes having state-dependent characteristics. Some of these factors are neglected in engineering practise for simplicity reasons.
In analysis of heat losses in pipelines, numerical methods are frequently adopted to solve the fluid flow and heat transfer problems in pipelines separately [1-5]. A finite element formulation for temperature-independent fluid flow in piping systems is proposed in [1], while heat flow analysis in a pipe as an axisymmetric insulated cylinder is illustrated in [2]. Moreover formulae for the heat loss from a pipe with insulation and a ground surface thermal resistance are proposed in [3]. Detailed evaluation of heat losses by the finite element method with attention on heat flux distribution into the soil has been solved in [4] and transient analysis of a piping system is presented in [5]. The emphasis of this report was to develop a general finite element approach in modelling of thermo-hydraulic flow for analysis of heat losses in a multilayered structure of a pipeline.
Finite element approach to coupled thermo-hydraulic flow in pipe
Generally the fluid flow in a pipe and heat transfer involving conduction, convection and radiation is a simultaneous phenomenon. The finite element approach to this coupled steady-state non-linear problem may be written in the following form:
(1) Here, [P] and [T] present nodal pressure and temperature vectors which have to be obtained by
Thermal
Fig. 1. Common insulated pipe
solution of the problem (1). [W] and [Q] describe given nodal fluid and heat flow vectors reflecting given boundary conditions, while [Khyd] and [Kther] present hydraulic and thermal matrices reflecting physical properties of fluid and insulation.
The fluid physical properties, which are involved in both heat transfer and fluid flow analysis, depend on temperature and this aspect is important in design calculations.
In other words, fluid flow and heat transfer analyses are coupled due to variation of the above properties, where the coupling is built into the governing equations implicitly. The solution of model (1) requires an iterative sequential procedure.
The hydraulic matrix mainly depends on the fluid including fluid-pipe interaction properties.
The physical nature of the thermal matrix is much more complicated. For an individual element e it reflects heat transfer through a multilayered, or even multisolid, structure of the pipe and environment and has the form of:
∑ ∑
Here, subscript m denotes all layers of insulation and environment structure producing thermal resistances. On the other hand, individual layers may contain different transfer modes. Each layer may produce a thermal matrix:
[Km]=[K1(h)+K2+K3+K4] (3)
Here the [K1] matrix describes convection to the layer adjacent to a surface through the convection coefficient h, [K2] reflects mass transport, while [K3] and [K4] describe conduction occurring in the flow longitudinal direction and normal to the flow direction, respectively.
For the fluid layer, the thermal matrix has the form: [Km]=[K1(h)+K2+K3]. For describing conduction in the insulation, the following matrix is used: [Km]=[K4]. For designing the combined effects of thermal radiation and natural convection from insulation surface, the thermal matrix has form of: [Km]=[K1(h)].
Evaluation of heat losses provides the same sequence as thermal matrices. For the individual element e:
Implementation of the above scheme requires development of a complex thermal element. The model (1)-(3) can be applied to the solution with different level of coupling the thermal characteristics.
Solution of heat losses in multilayered structure of the heating pipe
The most common pipes in use for district heating is insulated pipes with carrier of steel, insulation of poly-urethane and a casing of high density polyethylene.
(Fig.1) In this case, the heat transfer from fluid to environment is a three-step process: from a warmer fluid to a wall, through the multilayered wall, then to a colder surrounding air.
The main problem occurring in a thermo-hydraulic finite element analysis of the pipe is that the existing finite element codes usually contain thermo-hydraulic pipe
Heat losses
D20 D40 D80 D160
Diameters of the pipe [mm]
Fig.4 The comparison of the heat losses between the finite element results (FEM) and results by an analytical method
element paying no attention to the structure of the pipe. In the framework of this paper the complex thermal element designing myltilayered structure of insulation has been proposed. It is compatible with the fluid element of the ANSYS code [6].
K
Fig.2 Complex thermal finite element and fluid element
In the present study the analysis of heat losses is based on the following assumptions:
1. The heat transfer process from fluid to surrounding is only solved in the radial direction, therefore it presents a one-dimensional (axi-symmetric) problem.
2. The pipe is assumed to be placed in an underground culvert and the temperature of surrounding air in the culvert is assumed to be known.
The developed complex thermal element contains temperature-dependent thermal properties, while evaluation of heat losses is
element
Fig.3 Complex thermal finite element
developed as separate postprocessor. A general scheme of the fluid element connecting pipe nodes I-J with the thermal element is presented in Fig. 2. Thermal elements K-I and L-J describe structure of insulation and environment as shown, in Fig.3. Here, heat transfer from fluid to wall describes fluid element, including flow - dependent convection coefficient. A set of conduction elements is needed to present the transfer of heat in the pipe, insulation and casing material, since properties of each material are different.
Heat transfer from the wall is represented by a convection element, which involves combined effects of natural convection and thermal radiation (Fig 3.)
Numerical example
A numerical example is applied to present the idea discussed in the previous section. Numerical examples are carried out for different diameters of the pipe (D20, D40, D80, D160 mm) and for a pipe length of 1 m. The temperature of the water is assumed to be 80oC, temperature of the surrounding air is 30oC. The properties of water are considered as follows: heat conductivity 0.684 [W/mK], specific heat 4313[J/(kgK)], density 993-937.2 kg/m3, viscosity 6.2-2.19×10-4 [kg/ms]. The heat conductivity of steel pipe is 76 [W/mK], while heat conductivity of insulation and casing is 0.032 and 0.43 [W/mK], respectively.
Fig. 4 shows the comparison of the losses between the finite element results and results by an analytical method for uncoupled problems.
The influence of coupling by assuming a temperature dependent variation of density and viscosity is also investigated. The proposed problem is also tested by the analysis of a fragment of a heating network.
Accuracy of Finite Element Model
The accuracy of the finite element model for the estimation of the heat losses is examined for a few test examples by using different level of discretisation. The first test example defines the influence of discretisation of conduction element. The relative accuracy of heat flow decreases when the insulation thickness decreases, but remains independent on thermal conductivity of the insulation material. The second test defines the influence of discretisation of fluid element.
Recommendations for longitudinal subdivision of the pipe section into fluid element are proposed as well.
Results and discussion
The finite element approach for thermo-hydraulic problem of a pipeline, mainly with a multilayered structure, with emphasis on analysis of heat losses has been developed. Theoretical investigation and numerical examples prove suitability of the above approach in engineering and advantages in comparison to the engineering method:
1. Multilayered structures of the pipe and surroundings are considered in the analysis of the heat losses.
2. Heat transfer modes at different level of coupling with the fluid flow by describing temperature dependent fluid characteristics can be investigated.
3. The finite element model with different accuracy may be used to estimate heat losses.
However, the numerical examples are limited because real heating pipes are placed in underground culverts.
Acknowledgement
Financial support for this investigation by the Nordic Energy Research Programme is gratefully acknowledged.
REFERENCES
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