ZHOU Shou-jun (周守军)

School of Thermal Energy Engineering, Shandong Jianzhu University, Jinan 250101, China,

E-mail:zsjun7342@sina.com

TIAN Mao-cheng (田茂诚)

School of Energy and Power Engineering, Shandong University, Jinan 250061, China

ZHAO You-en (赵有恩)

Department of Computer Science and Technology, Shandong University of Finance and Economics, Jinan 250014, China

GUO Min (郭敏)

School of Thermal Energy Engineering, Shandong Jianzhu University, Jinan 250101, China

Dynamic modeling of thermal conditions for hot-water district-heating networks*

ZHOU Shou-jun (周守军)

School of Thermal Energy Engineering, Shandong Jianzhu University, Jinan 250101, China,

E-mail:zsjun7342@sina.com

TIAN Mao-cheng (田茂诚)

School of Energy and Power Engineering, Shandong University, Jinan 250061, China

ZHAO You-en (赵有恩)

Department of Computer Science and Technology, Shandong University of Finance and Economics, Jinan 250014, China

GUO Min (郭敏)

School of Thermal Energy Engineering, Shandong Jianzhu University, Jinan 250101, China

(Received March 23, 2013, Revised January 6, 2014)

To investigate the dynamic characteristics of the thermal conditions of hot-water district-heating networks, a dynamic modeling method is proposed with consideration of the heat dissipations in pipes and the characteristic line method is adopted to solve it. Besides, the influences of different errors, space steps and initial values on the convergence of the dynamic model results are analyzed for a model network. Finally, a part of a certain city district-heating system is simulated and the results are compared with the actual operation data in half an hour from 6 secondary heat stations. The results indicate that the relative errors for the supply pressure and temperature in 5 stations are all within 2%, except in one station, where the relative error approaches 4%. So the proposed model and algorithm are validated.

district-heating network, thermal conditions, dynamic modeling, characteristic line method

Introduction

With decades of the rapid economical development and urbanization in China, the district-heating is considerably developed. The heating scale is expanding, and the pipe networks become more and more complex[1]. As a result, the dynamic transition process during the operation regulation takes ever longer time, and its impact on the system operation becomes an important issue. Therefore, in order to realize a hydraulic and thermal balance and to provide the desired heating load for a district-heating system, the dynamic operation mechanism and characteristics of the district-heating network must be researched.

So far, there are mainly two ways to investigate the dynamic operating characteristics of a district-heating network: the system identification method and the method based on the physical model law. The former is simpler because it does not concern the operating mechanism of the identification object, and it concentrates only on the relationship between the input and output parameters of the measured data. However, it also has many drawbacks: (1) Poor generalization ability. Once the structure of a pipe network changes, a new model is needed, to be obtained by analyzing the measured data of the new system. Thus this model can not be applied to continuously developing and expanding heating networks. (2) The influence of a network topology on the operation conditions is not considered, thus the operation mechanism of a network can not be obtained. This problem becomes more serious when the network is large. The latter method involves onerous modeling work because it deals withthe entire heating system, including the heat source, the heating network, and the heat users. However, the drawbacks of the former method are overcome. The related studies can be found in Denmark, Germany and other countries. Benny Bohm and his team proposed a node method to study the aggregated simulation model of the pipe network[2-5]. A. Loewen studied the simulation system and the structural simplification of the complex network to save the calculation time[6,7]. Some studies can also be found in China[8,9].

1. The dynamic model of a hot-water district-heating network

The flow in a hot-water heating network can be regarded as one-dimensional and incompressible. The pressure wave propagates in water at the sound speed, around 1 200 m/s, while the temperature wave spreads at the speed close to the flow rate in the pipes. Thus the pressure-wave transmission is about 1 000 times faster than that of the temperature wave in a heating network. So, it can be assumed that the flow becomes stable instantaneously. And we can set up the dynamic model of the thermal conditions based on the current steady flow. Finally, a quasi-dynamic model of a hotwater district-heating network is thus established.

1.1 The thermal dynamic model of a district-heating system

The plane network equations are used in the steady hydraulic model[10]. Its thermal conditions can further be simplified as follows: (1) the axial thermal conduction of the fluid and the wall in a pipe can be ignored because they are very small compared with the radial heat conduction, (2) the heat dissipation can be ignored because it has little effect on the fluid temperature under the low velocity condition, (3) the shear stress is the same as that of the steady flow and can be calculated by the Darcy-Weisbach formula[11].

According to the mass conservation and the momentum conservation for one-dimensional incompressible fluid, we have

where C is an arbitrary constant,qvis the volume flux,λis the fiction resistance coefficient,ρis the fluid density,Dis the pipe diameter,θis the angle between the pipe and the horizontal direction.

Based on the energy conservation, we have

whereq is the increased heat in the control volume per unit surface area and per unit time, the heat emission is positive and the heat absorption is negative, W.

Because the constant-pressure specific heat of the hot-water changes little with the change of the fluid temperature and pressure, it can be regarded as constant . Thus the flui d enthalpy is only influ ence d by the fluidtemperature.SoEq.(10)representsthethermaldynamic model for one-dimensional incompressible flow in a pipeline. It is a first-order linear hyperbolic equation with constant coefficients.

1.2 The heat-loss equation of directly buried hotwater pipelines

Because the temperature of the undisturbed soil changes very slowly and has a certain time lag as the average temperature of the outdoor-air changes during the whole year, the variation of the pipe heat-loss mainly depends on the temperature change of the fluid in a pipe during a short term.

Therefore, the method of least squares is used to obtain the heat-loss equations of the supply and return pipelines for different diameters and different insulation materials. The equations take the hot-water temperature as the variable. Then these equations can be used by the thermal dynamic model of the heating network[13].

2. The solving algorithm for the thermal dynamic model

2.1 Establishing the corresponding characteristic equation

The characteristic method can translate a hyperbolic quasi-linear partial differential equation into an ordinary differential equation along the characteristic lines, and then its numerical solution can be obtained. So, we have

When the current flux is constant, the corresponding characteristic line is a straight line. Substituting Eq.(11) into Eq.(10), we have

Eqation (12) is the characteristic equation for the thermal dynamic model of a district-heating network.

2.2 Establishing the characteristic differential equation

The characteristic line method for one-dimensional unsteady flow problems includes a variety of finite difference grids and stepping algorithms. The inverse step method is selected in this paper[14]. To ensure the stability of the solution, the time step must meet the Courant-Friedrichs-Lewy (CFL) condition[15].

Expanding Eq.(12) by first-order Tailor series and a forward explicit difference scheme, it becomes a finite difference equation along the characteristic line

With further simplifications, it becomes

where m is the node number,nis the index of the time layer.

3. Thermal dynamic modeling of the primary network

3.1 The primary network

To analyze the proposed model and algorithm, we establish a primary heating network, which includes 10 secondary stations.

The main features of a district-heating network are included with some details reasonably simplified to facilitate the modeling and the dynamic characteristic analyzing, as shown in Fig.1.

3.2 Analyzing the thermal dynamic model of the primary network

The intermediate user (U5) and the terminal user (U10) are selected as the main studying objects. 6 key nodes are identified, including the entrance node and the exit node of the two users, the supply node and the return node of the heat source, to analyze the temperature dynamic characteristics of the transition process from a disturbed state to the next stable state of the network, as shown in Fig.2.

The simulated condition C1: assuming that the outdoor-air temperature rises, and the heat load of each user reduces to half of its designed heat load. Tohe supply temperature of the heat source is set to 120C. The program iterates 370 times to convergence.

It can be seen from Fig.2 that the supply temperatures of the heat source and each user tend to rise under the influence of the initial temperature profile since the simulated conditions are identical to the designed conditions. The delay time of each user is related with its distance away from the heat source. Afterwards, the inlet temperatures of users keep stable.

The outlet temperatures of the users have an obvious jump from the initial value (at the first node) since the heat load is reduced by 50%. Then they rise slowly and become stable under the influence of the initial temperatures. The varying trends of the inlet and outlet temperatures are the same for each user. The backwater temperature of the heat source changes most greatly.

3.3 Influence of different errorsεon simulation results

Under the condition C1, we compare the simulation results with different errors (0.1, 0.01 and 0.001). Comparing Fig.3 with Fig.2, we can see that the inlet and outlet temperatures of the users tend to be steady, and the final temperatures of various nodes in Fig.3 are almost the same as the corresponding ones in Fig.2. The heat-source backwater temperature in Fig.3 can not reach the final steady state in Fig.2. As a result, the final heat-source backwater temperatures are quite different in the two figures.

Comparing Fig.4 with Fig.2, we can see that the temperature difference is the greatest at the node of the heat-source backwater temperature, about 1oC. The iterative count increases by 212, and the transient transition process extends by 1.73 h.

3.4 Influence of different step sizes h on simulation results

In order to ensure that the final state is identical for different step sizes, we must choose the temperature gradient as the criterion for convergence, which is the ratio of the temperature difference between two iterations to the corresponding time step.

Under the condition C1, the results for different step sizes are shown in Fig.5 and Fig.6. The iteration counts are 603 and 788, respectively. From the two figures, we can see that the varying trend of temperatures and the final state of each node are consistent. Therefore, we can choose the larger step size (50 m) to save the calculation time.

3.5 Influence of initial value on convergence

The designed operation condition of a districtheating system is easily available, so it is reasonable to take it as the initial operation condition of the program. If the operation condition of any time can be obtained by the district-heating automatic monitoring system and used to be the initial operation condition in the actual operation, it would be much easier to reach the ultimate convergence state compared with the designed condition. So, the designed condition may be the worst initial condition than any actual condition.

4. Validation of dynamic model and algorithm for district-heating networks

We take the district-heating system of Linyi city, Shandong province as the research object, and focus on its eastern trunk heating network (known as the East Route).

4.1 The topology and corresponding network parameters of the East Route

The main structural parameters of the East Route are shown in Fig.7. An on-line monitoring system is installed, to monitor mainly the temperatures and the pressures of the supply and return water for the primary and secondary sides, and the operation of the pumps at each secondary heat station, also the local outdoor-air temperature. The monitoring system issues reports once per minute.

The heating pipes are mainly directly buried in Linyi network, some pipes across rivers are overhead and few pipes are laid by trench. The prefabricated insulation pipes are chosen for the buried pipes, with the high-temperature rigid polyurethane as insulation materials and the high density polyethylene pipe is used for the pipes in open places.

The actual heating area in the 2008-2009 heating season of Linyi heating network is 5.797×106m2. The heating area of the East Route is 1.7495×106m2, which is nearly one-third of the total heating area.

4.2 The validity and accuracy of the model and algorithm

The thermal model of the East Route is established by the above mentioned method. The supplytemperature and pressure at 6 typical secondary heat stations from 8:00 to 8:30 at 12 December, 2008 are used as the validation data. According to the records, the average circulation flux is 5484 t/h, the average lift of the circulation pump is 135 m H2O, the supply temperature of the heat source is 93.5oC, and the average out-door-air temperature is 5oC.

The measured and simulated supply temperatures and pressures for each typical station are shown in Table 1 and Table 2.

A certain deviation can be seen between the measured and the simulated data, due to three reasons: (1) the simulation model needs a lot of network parameters for the pipeline structure, the pipe roughness, the pipe insulation, the valve type and the valve location and so on. And some parameters will change during the running of the actual network. Sometimes, it is hard to determine the accurate network data, (2) some actual pipelines are laid by trench or overhead, but the simulation model deals only with a direct burial, (3) the measured data have instrumental error since the supply temperature and pressure on the primary side are obtained by the field thermal resistance and the remote transmitting pressure gauge.

5. Conclusion

Based on reasonable simplifications, the thermal dynamic model and its solution algorithm for a hotwater district-heating system are developed, and the many influencing factors, including the initial value, the error and the space step are analyzed. The East Route in Linyi heating network is selected as the simulation object. The monitoring data of the supply temperature and pressure on the primary side in half an hour at 6 typical stations are selected as the verifying data. It is found that the relative error of the supply temperature and pressure at 5 stations are all within 2%. For the 13th station, the relative errors are 3.07% and -3.95%, respectively. Thus the validity and accuracy of the proposed model and algorithm are established. In this paper, the tree-like city hot-water district-heating system of a single heat source is simulated and analyzed. This study will provide some food for thought for further research of large and complex heating networks, such as the networks with multiple heat sources, the annular pipe networks and so on.

Acknowledgement

This work was supported by the Doctoral Scientific Research Fund Program of Shandong Jianzhu University (Grant No. XNBS1225), the School Scientific Research Fund Program of Shandong Jianzhu University (Grant No. XN110108), the Key Laboratory of Renewable Energy Utilization Technologies in Buildings, Ministry of Education and the Key Laboratory of Renewable Energy Application Technologies in Buildings, Shandong Province.

[1] JIANG Yi. Problems in improvement of central heating systems in China and possible solutions[J]. Journal of Heating, Ventilating and Air Conditioning, 2006, 36(3): 37-41(in Chinese).

[2] GARIELAITIENE I., BOHM B. and SUNDEN B. Modelling temperature dynamics of a district heating system in Naestved, Denmark–A case study[J]. Energy Conversition and Management, 2007, 48(1): 78-86.

[3] GARIELAITIENE I., BOHM B. and SUNDEN B. Evaluation of approaches for modeling temperature wave propagation in district heating pipelines[J]. Heat Transfer Engineering, 2008, 29(1): 45-56.

[4] LARSEN H. V., BOHM B. and WIGBELS M. A comparison of aggregated models for simulation and operational optimization of district heating networks[J]. Energy Conversion and Management, 2004, 45(6): 1119-1139.

[5] LARSEN H. V., PALSSON H. and BOHM B. et al. Aggregated dynamic simulation model of district heating networks[J]. Energy Conversion and Management, 2002, 43(8): 995-1019.

[6] LOEWEN A., WIGBELS M. and ALTHAUS W. et al. Structural simplification of complex DH-networks[J]. Euroheat and Power/Fernwarme internation, 2001,30(5): 42-44.

[7] LOEWEN A., WIGBELS M. and ALTHAUS W et al. Structural simplification of complex DH-networks-Part 2[J]. Euroheat and Power/Fernwarme internation, 2001, 30(6): 46-50.

[8] ZHOU Xue-ling, JIANG Yong-cheng and LI Feng. Thermal condition model of the central heating network[J]. Journal of Harbin Institute of Technology, 2005, 37(12): 1683-1685(in Chinese).

[9] WANG Si-ying, ZOU Ping-hua and ZHOU Zhi-gang et al. Thermal condition calculation model of direct connection hot water heating system based on graph theory[J]. Journal of Heating, Ventilating and Air Conditioning, 2011, 41(8): 106-109(in Chinese).

[10] ZHOU Shou-jun, ZHAO You-en and CHEN Ming-jiu et al. Research of the hydraulic adjustment method for a district heating system[J]. Journal of Shandong University (Engineering Science), 2009, 39(3): 151-153, 158(in Chinese).

[11] WU Chi-gong. Hydraulics[M]. Forth Edition, Beijing, China: Higher Education Press, 2008, 81-86(in Chinese).

[12] WANG Yong-zhen, CHEN Gui-tang. Advanced engineering thermodynamics[M]. Beijing, China: Tsinghua University Press, 2013, 75-76(in Chinese).

[13] ZHOU Shou-jun. Operational parameter prediction and optimization research of district heating system based on pipe network dynamic model[D]. Doctoral Thesis, Jinan, China: Shandong University, 2012(in Chinese).

[14] TANG Jian-feng, DUAN Chang-gui and LV Wen-zhe. The application of characteristic line method for dynamic simulation of a gas pipeline[J]. Oil and Gas Storage and Transportation, 2001, 20(8): 12-17(in Chinese).

[15] LU Jin-pu, GUAN Ye. Numerical solution of partial differential equations[M]. Second Edition, Beijing, China: Tsinghua University Press, 2004, 13-36, 49-53(in Chinese).

10.1016/S1001-6058(14)60060-3

* Project supported by the Scientific Development Program of Shandong Province (Grant No. 2012GGB01071).

Biography: ZHOU Shou-jun (1974-), Male, Ph. D.,

Associate Professor