Abstract

In this study, we analyze the deficiencies of specific frictional resistance in heating engineering. Based on economic specific frictional resistance, we put forward the concept of comprehensive specific frictional resistance, which considers the multiple factors of technology, economy, regulation modes, pipe segment differences, and medium pressure. Then, we establish a mathematical model of a heating network across its lifespan in order to develop a method for determining the comprehensive specific frictional resistance. Relevant conclusions can be drawn from the results. As an application, we have planned the heating engineering for Yangyuan County in China, which demonstrates the feasibility and superiority of the method.

1. Introduction

With the development of modern society, improvements in heating engineering have attracted attention. In the field of heating engineering, the specific frictional resistance refers to the resistance loss of pipe per meter [1, 2], which is very important in China. In this paper, we discuss specific frictional resistance during the designing or planning stage, when specific frictional resistance acts as an intermediary to select a suitable pipe diameter.

At present, studies mainly involve two aspects of specific frictional resistance: one is the weighted average specific frictional resistance; the other is economic specific frictional resistance. For a heating network with regulated change in flow rate at different stages, the specific frictional resistance undergoes a step change due to the change in flow rate. In order to measure and reflect this characteristic of the specific frictional resistance, weighted average specific frictional resistance is recommended for selecting pipe diameters. Hong [3] discussed the differences between weighted average frictional resistance and traditional specific frictional resistance and analyzed the power-saving effect by taking a heating network with flow rate regulation as an example. The economic specific frictional resistance is an ideal value relative to the specific frictional resistance from the economic perspective and is widely used in China at present. Ge et al. [4] established a mathematical model, which took the minimum cost of the heating network as the objective function, and discussed the relationship between the economic specific frictional resistance and the economic heating radius. Liu et al. [5] offered the concept of the new economic specific frictional resistance by establishing the transportation distance limit model and the transport energy consumption ratio model. Wang et al. [6] took the indoor heating horizontal single tube and vertical single tube system as examples to establish a mathematical model with minimum cost as the objective function, by which they obtained the economic specific frictional resistance under different conditions. Li et al. [7] studied the hydraulic calculation methods and the existing problems on the condensate pipe net, and they used the differential method to calculate the economic specific frictional resistance of the condensate pipe net.

Through analysis of the various research [3–11] on specific frictional resistance, we identified problems related to the value of specific frictional resistance, as follows:

(i) In China, the value of specific frictional resistance used in engineering still follows the traditional economic specific frictional resistance value of 30~70 Pa/m, which was introduced by the former Soviet Union in the last century. With the rise in prices and changes in materials, this value is not applicable to the current state of practice in heating engineering.

(ii) The value of specific frictional resistance should be related to the regulation modes, which can bring about different changes in flow rate and medium temperature. However, the traditional model of economic specific frictional resistance treats the different regulation modes in the same way [3–11].

(iii) With the development of the central heating network, pipe networks have become more massive and complex, which has led to increasing differences among pipe segments. Thus, it is not suitable to use the same specific frictional resistance standard of 30~70 Pa/m to determine the pipe diameter in a network system. Instead, each pipe segment should be differentiated. That is to say, each pipe segment should have a specific frictional resistance of its own [3–11].

(iv) The distribution of pressure in the pipes is now more complicated because of the mass and complexity of the pipe networks. It has a direct relationship to the value of specific frictional resistance whether the pressure meets the requirements. However, the value of traditional economic specific frictional resistance of 30~70 Pa/m does not take into account the constraints of the medium pressure [3–11].

In order to solve the above problems, we propose the concept of comprehensive specific frictional resistance. It refers to a measure of resistance loss of pipe per meter that incorporates a number of related factors. By means of it, heating engineering applications can achieve better economic effects during their lifespan [12–14]. The related factors considered in this paper include technology, economy [15, 16], operation regulation modes [17, 18], pipe segment differences, and the medium pressure [19–22].

2. The Mathematical Model

In this section, we give the method for obtaining the value of comprehensive specific friction resistance as follows.

2.1. Factors Related to Comprehensive Specific Frictional Resistance Theory

Through the analysis of technology, economy, operation regulation modes, pipe segment differences, and the medium pressure in a heating system, we have drawn several conclusions about the comprehensive specific friction resistance model theory:

(i) Pipe Segment Differences. With pipe networks becoming massive and complex, the difference among pipe segments becomes more apparent. For example, it is common for the pipe diameter close to the heat source to be DN1400, while the pipe diameter can be reduced to DN200 at the user end point. This difference should not be ignored. In order to account for the differences among pipe segments, a single pipe segment is taken as an object in establishing the mathematical model in this paper. In this way, we get different values corresponding to different pipe segments.

(ii) Operation Regulation Modes. We take into account the influence of operation regulation modes on comprehensive specific frictional resistance due to the fact that different operation regulation modes incur different electricity consumption and different heat loss.

(iii) Medium Pressure. The value of comprehensive specific frictional resistance takes into account the medium pressure constraint, which can prevent heating accidents on the operational stage, such as pipe cracking or medium vaporization. In the following study, the medium pressure constraint is mainly embodied in the constraint conditions of the mathematical model.

(iv) Technology and Economy. The Present Value Cost is chosen as the objective function [23, 24] by analyzing the advantages and disadvantages of each economic effect evaluation index.

(v) Technology and Economy. The value of specific frictional resistance is sensitive to the factors of primary network construction fee (including civil engineering cost and pipe installation fee), electricity fee, and fuel cost across the lifespan of heating engineering application. These factors must be embodied in the mathematical model. Heat source construction fee, heat exchange station construction fee, thermal user fuel cost, water fee, maintenance cost, and labor wages are irrelevant factors that can be ignored in the mathematical model.

2.2. Establishment of the Objective Function

In this section, we establish the objective function taking the single segment as the object. The objective function is the Present Value Cost of pipe segment across the lifespan of years, and the cost includes civil engineering cost, pipe installation fee, electricity fee, and fuel cost.

For the different regulations, the civil engineering cost and the pipe installation fee are the same, but the electricity fee and fuel cost are different, because the heat load varies continuously during heating season. The following is the establishment process of the objective function.

2.2.1. The Civil Engineering Cost

Taking the directly buried laying method as an example, a set of data about pipe diameter and the corresponding civil engineering cost from Chinese markets is shown in Figure 1.

Figure 1 

Fitting curve of heating pipe diameter and civil engineering cost.

The fitted curve is , whose -squared value is as high as 99.95%. Hence, the relationship between the civil engineering cost and the pipe diameter can be expressed by a quadratic polynomial, as follows:where is the civil engineering cost of pipe with a diameter (Ұ/m), is the pipe diameter (m), and , , and are the fitting polynomial coefficients of civil engineering cost, which are determined by the specific situation.

As we all know, the Darcy-Weisbach Formula iswhere is the head loss (m), is the coefficient of frictional resistance [25–27], is the length of pipe (m), is the velocity of medium in the pipe (m/s), and is the gravitational acceleration (m/s²).

The specific frictional resistance refers to the resistance loss of pipe per meter, so we get formula (3) from formula (2):where ρ is the density of medium (kg/m³) and is the specific frictional resistance (Pa/m).

Therefore, the specific frictional resistance, pipe diameter, and flow rate have the following relationship:where is the flow rate (kg/s).

For a heating network, the determination of the comprehensive specific frictional resistance needs to account for the change in flow rate during operation. However, the selection of the diameter through the flow rate and the specific frictional resistance still relates to the most unfavorable situation. Therefore, the flow rate in formula (4) is the design flow rate . That is,where is the pipe diameter of pipe segment (m), is the design flow rate of pipe segment (kg/s), and is the specific frictional resistance of pipe segment (Pa/m).

Applying the basic formula of specific heat capacity ( is the specific heat capacity, which for water is 4.2 × 10³ J/(kg·°C), m is the quality (kg), is the temperature difference (°C), and is the heat quantity (J)) to heating engineering, the following formula can be obtained:where is the thermal load (W), is the temperature difference (°C), and is the flow rate of medium (kg/s).

Substituting formula (6) into formula (5), we getwhere , is the design temperature difference between supply and return water of pipe segment (°C), and is the design thermal load of pipe segment (w).

Substituting formula (7) into formula (1), the civil engineering cost of pipe segment can be given bywhere is the civil engineering cost of pipe segment (Ұ) and is the length of pipe segment (m).

2.2.2. The Pipe Installation Fee

The data from Chinese markets about pipe diameter and the corresponding pipe installation fee is shown in Figure 2.

Figure 2 

Fitting curve of heating pipe diameter and pipe installation fee.

The fitted curve is , whose -squared value is as high as 99.73%. Hence, the relationship between the pipe installation fee and the pipe diameter can also be expressed by a quadratic polynomial, as follows:where is the installation fee of pipe with a diameter (Ұ/m) and , , and are the fitting polynomial coefficients of the installation fee, which are determined by the specific situation.

Substituting formula (7) into formula (9), the installation fee of the pipe segment can be given bywhere is the installation fee of the pipe segment (Ұ).

2.2.3. The Electricity Fee

(i) For a Heating System with Quality Regulation. A heating system with quality regulation is regulated simply by changing the water-supply temperature, so the flow rate of pipe segment is always the design flow rate . Because the specific frictional resistance R refers to the resistance loss of pipe per meter, the annual electricity consumption of pipe segment iswhere is the annual electricity consumption of pipe segment (J) and is the run time of pipe segment per year (s).

So, the annual electricity fee for the pipe segment can be given bywhere is the annual electricity fee for pipe segment (Ұ), is the electrovalence (Ұ/J), and is the pump efficiency.

Substituting formula (6) into formula (12), we get

(ii) For a Heating System with Flow Rate Regulation. A heating system with quality regulation is regulated by changing the flow rate. The large variation on the thermal load during the entire heating period results in a wide range of flow rate changes, so calculating the electricity fee according to the design flow rate does not adequately describe the actual situation. In order to estimate the electricity consumption accurately, we need to know the local thermal load changes and the corresponding hours to draw a thermal load duration diagram [28]. Under common conditions, the thermal load duration of the pipe segment in heating engineering can be described as Figure 3.

Figure 3 

Thermal load duration diagram of the pipe segment .

According to the definition of specific frictional resistance , the annual electricity consumption of pipe segment iswhere is the actual flow rate of pipe segment (kg/s), is the flow rate (kg/s)-duration (s) function, and is the duration corresponding to the minimum thermal load during a heating season (s).

The annual electricity fee for the pipe segment is

Substituting formula (6) into formula (15), the annual electricity fee for the pipe segment can be given bywhere , is the actual thermal load of pipe segment (w) and is the thermal load (w) – duration (s) function, as shown in Figure 3.

2.2.4. The Fuel Cost

The fuel cost includes two parts: one is heat loss fuel cost; the other is thermal user fuel cost. Due to the irrelevance of the specific frictional resistance to thermal user fuel cost, there is no need to embody it in the mathematical model. However, massive, complicated pipeline networks increase heat loss along the route, so heat loss fuel cost should be included in the total cost rather than ignored. The heat loss fuel cost is different for different regulation modes, because it is associated with the medium temperature.

(i) For a Heating System with Flow Rate Regulation. The heat transfer coefficient refers to the amount of heat transfer per unit time per unit surface area for a 1°C temperature difference, so for a heating network adopting the flow rate regulation mode, the consumption of heat loss fuel iswhere is the consumption of heat loss fuel (J), is the heat transfer coefficient of the pipe surface (w/m²·°C), is the ambient temperature of pipe segment (°C), and is the design temperature of medium in pipe segment (°C). For the water-supply pipe, , and for the water-return pipe, , , where is the design water-supply temperature of pipe segment (°C), is the design water-return temperature of pipe segment (°C).

So the annual heat loss fuel cost iswhere is the annual heat loss fuel cost of the pipe segment (Ұ) and is the fuel price (Ұ/J).

Substituting formula (7) into formula (18), the annual heat loss fuel cost of the pipe segment can be given by

(ii) For a Heating System with Quality Regulation. For heat-supply networks adopting the quality regulation mode, calculating the heat loss fuel cost by the design temperature is unreasonable due to the variance of the medium temperature in the pipe. As we know, compared with the design water-supply temperature and design water-return temperature, a reduction in actual water-supply temperature of Δ1 is equal to an increase of Δ2 of the actual water-return temperature in the quality regulation networks [8]. This is described in Figure 4.

Figure 4 

Duration of actual water-supply or water-return temperature.

So, the following formula can be given:where is the actual temperature of medium in pipe segment (°C) and is the actual temperature difference between supply and return water of pipe segment (°C), as shown in Figure 4.

According to the definition of heat transfer coefficient , the consumption of heat loss fuel is

Substituting formula (20) into formula (21), we get the consumption of heat loss fuel:

So, the annual heat loss fuel cost of the pipe segment is

Substituting formula (7) into formula (23), the annual heat loss fuel cost of the pipe segment can be given by

For a heating system with quality regulation, the actual flow rate is always the design flow rate , so we know

Substituting formula (25) into formula (24), we get

2.2.5. The Objective Function

For pipe segment , the objective function is the Present Value Cost in its lifespan of years. Supposing the discount rate is , it can be described as follows.

(i) For a Heating System with Quality Regulation

(ii) For a Heating System with Flow Rate Regulationwhere PC is the Present Value Cost (Ұ), is the discount rate, and is the life period (years), , , , .

2.3. Establishment of the Constraint Conditions

2.3.1. The Constraint Conditions of Medium Pressure

The pressure in the pipe must meet certain requirements in order to ensure normal operation of the system and prevent pipe burst or medium vaporization. As the research object is the primary network system, the lower pressure limit must guarantee that high-temperature water does not vaporize in the pipeline, while the upper limit is the pressure that the pipe and pipe fittings can bear. Additionally, the water-return temperature is generally less than 100°C, and for any water-supply pipe segment , there must be a corresponding water-return pipe segment whose pressure is lower than the pressure in the pipe segment . Therefore, only the water-supply pipes need to obey the pressure constraints. In addition, in order to ensure that the pressure in the pipe segment meets the requirements, the starting and ending points of the pipe must also meet the requirements.

A branched network diagram without distributed pressure pumps is shown in Figure 5. Pipe segments constitute the trunk line. Under normal circumstances, the constant pressure point is located at the entrance of the circulating pumps, that is, the end of pipe segment 1′. Assuming that the pressure at the constant pressure point is , the pressure at any section of any pipe segment can be calculated.

Figure 5 

A branched network diagram.

According to the Bernoulli equation (Bernoulli’s principle), the pressure at the start of pipe segment can be given by

So the pressure constraint condition of pipe segment can be given bywhere is the pressure at the start of pipe segment (Pa), is the flow velocity of the medium in pipe segment (m/s), is the topographic height at the start of pipe segment (m), is the pressure at the end of pipe segment (the constant pressure point) (Pa), is the flow velocity of the medium in pipe segment (m/s), is the topographic height at the end of pipe segment (m), is the resistance loss at heat exchange station (Pa), is the ratio of the local resistance to the resistance along the path, is the saturation steam pressure corresponding to the design water-supply temperature (Pa), is the maximum pressure that the pipe and pipe fittings can bear (Pa), and is the volumetric weight (N/m³), .

In order to simplify formula (30), the item can be moved as follows:

There is little difference among different pipelines in terms of flow velocity, for which the range is 0~4 m/s [29, 30]; that is, the value of is −8000~8000 Pa. Compared with and , its order of magnitude is low. Therefore, formula (31) can be simplified to the following form:

In this way, not only has the formula been simplified, but also a certain margin has been set for the upper and lower limits.

From the Antoine equation [31], we know that  Pa, so the pressure constraint condition at the start of pipe segment can be given bywhere .

As described above, the pressure at the end of pipe segment is equal to the pressure at the start of pipe segment because the velocity difference has been set as a margin for the upper and lower limits, which means the pressure constraint at the end of pipe segment has been validated in the previous step by the pressure constraint of pipe segment . Therefore, the pressure constraint at the end of pipe segment can be ignored.

2.3.2. The Constraint Condition of Design Temperature Difference between Supply and Return Water

The design temperature difference between supply and return water should be limited to a certain range. Too small a temperature difference can result in reduced and inadequate heat exchange, while too large a temperature difference may need a very high water-supply temperature, which may be difficult to achieve or may incur a very high cost in practice. Therefore, the design temperature difference between supply and return water should be limited by

2.3.3. The Constraint Condition of the Flow Velocity

The flow velocity in the pipes should be restricted to a certain range, as follows:

Substituting formulas (6) and (7) into formula (35), the constraint conditions of the flow velocity can be given by

2.4. Establishment of a Mathematical Model

Based on the above analysis, a mathematical model of Present Value Cost to determine the comprehensive specific frictional resistance of pipe segment can be given as follows.

(i) For a Heating System with Quality Regulation

(ii) For a Heating System with Flow Rate RegulationFor the water-supply pipes, the constraint conditions are the inequalities (39), (40), and (41), while, for the water-return pipes, the constraint conditions are the inequalities (39) and (40).

2.5. The Solvability Analysis of the Mathematical Model

The objective function PC(, ) we established is a continuous real valued function map, and the value ranges of the decision variable and are compact sets. Therefore, we can know by the Weierstrass Theorem [32] that there must be a solution vector (, ) that makes the objective function a minimum.

2.6. The Analysis of Algorithm

The mathematical model established above belongs to the two-dimensional extreme value problem under inequality constraints. The Complex method is selected as the optimization algorithm and is used to solve the models, whose function is to solve the -dimensional extremum problem under the equality and inequality constraints.

2.7. The Value of Comprehensive Specific Frictional Resistance

According to the actual engineering, the basic parameters in the objective function and the constraint conditions can be determined, while the total heat load of the heat consumers that the pipe segment provides heat for during the heating period can be calculated. Substitute all of them into the mathematical models (37)~(41) and solve the mathematical model. We can obtain the comprehensive specific frictional resistance of the pipe segment and the design temperature difference between supply and return water .

3. Application

3.1. Case

We have planned the heating engineering for Yangyuan County in China. The heating system is based on flow rate regulation as the main regulation mode, considering its energy-saving advantage. In the planning process of the network, the two schemes, comprehensive specific frictional resistance and economic specific frictional resistance, are used to optimize the pipe diameters.

According to market research, the calculation parameters have been selected as follows.

Coefficient of frictional resistance , discount rate , electrovalence  Ұ/J, fuel price  Ұ/J, the lifespan years, run time per year  s (equivalent to 168 days), pump efficiency , ambient temperature of pipe segments °C, heat transfer coefficient of pipe surface  w/(m²·°C), design water-return temperature °C, fitting polynomial coefficients of civil engineering cost , , and , fitting polynomial coefficients of pipe installation fee , , and , the ratio of local resistance to resistance along the path , the resistance loss at heat exchange station  MPa,  MPa, upper bound of flow velocity  m/s, lower bound of flow velocity  m/s, upper bound of design temperature difference between supply and return water °C, lower bound of design temperature difference between supply and return water °C, upper bound of pressure  MPa,  kg/m³, and  N/m³, and the duration of outdoor temperature in Zhangjiakou are shown in Table 1.

According to the site investigations, and to follow the trends of the existing pipeline network, the whole heating area was divided into three independent heating areas as shown in Figure 6: the planning floor area of heating area A is 9.61 × 10⁶ m², the planning floor area of heating area B is 6.57 × 10⁶ m², and the planning floor area of heating area C is 4.75 × 10⁶ m².

Figure 6 

The map of heating areas A, B, and C.

In this study, we take heating area C as an example. In Figure 7, only water-supply pipes are marked, while the position and trend of water-return pipes are in agreement with the corresponding water-supply pipes. In the following description, the numbers with “′” represent the water-return pipes, while the numbers without “′” represent the water-supply pipes.

Figure 7 

Heating network in heating area C.

The entire calculation process is as follows:

① Choose the most distant loop from the heat source (the trunk line), that is, ;