Entropy Flow Analysis of Thermal Transmission Process in Integrated Energy System Part I: Theoretical Approach Study

Abstract

It is very important to accurately describe the dynamic processes of thermal energy transmission for coupling with Integrated Energy System (IES). In order to study the thermodynamic characteristics of heat supply, this paper theoretically suggested a generalized model of entropy flow by deducing the expression of entropy conduction and convection based on thermodynamic law and heat transfer analysis. Taking temperature and entropy as the intensity and extension properties, the equivalent distributed and lumped parameter models are established to describe the features of heat loss and transmission delay. The effectiveness of current models is verified by comparing with solutions of conventional Partial Differential Equations (PDE) of heat transfer. The numerical simulation and verification procedure were conducted by Matlab/simulink. The proposed models were applied to simulate the response of temperature and entropy flow of a pipe with length of 100 m under different discrete conditions. The results show that for a distributed parameter model the maximum relative error is 1.275% when the pipe is divided into 100 sections, and for a lumped parameter model, the overall relative error is in the order of 10⁻³, which can be ignored in practical applications. All these prove the correctness of proposed models in this paper.

1. Introduction

The dynamic analysis of thermal energy system and thermal energy transmission is very important for Integrated Energy System (IES), which is directly related to the efficient and safe operation of IES [1,2,3,4]. The central heating network is the main form in which thermal energy plays a role in the integrated energy system, and it is also an important infrastructure to ensure the production and life of urban residents in winter [2,3]. Its stable operation is of great significance to the energy conservation and emission reduction of the heating system. The heating pipeline and its pipe network system is an important part of connecting the heat source and load users. Moreover, the structure of pipe network is the basis for building the dynamic performance of the heat supply network and the skeleton of thermal energy transmission. Therefore, it is of great significance to study the physical and dynamic thermal characteristics of central heating network for the optimal scheduling of heating network coupling with IES [2,3,4].

The research on thermal dynamic characteristics of heating network needs to comprehensively consider the heat loss and temperature delay in the process of heat transmission and distribution. There are two main ways to analyze thermal dynamic characteristics, one is the finite element method, and another is so-called node method [5,6,7,8]. The former is based on the discrete solution of the continuum by finite element method, which simplifies the complex problem and obtains the approximate solution of the focused finite element domain. The capacity of the calculation resource demand is directly related to the complexity and accuracy of solution to the problem. The latter is a finite difference method. The main principle is to use the temperature values of different time steps to describe the temperature distribution, and obtain the temperature values on the outflow node through the history value records on the inflow node [9]. Both finite element method and node method belong to the category of mechanism modeling, which has clear physical significance, and the modeling process is simple and easy to follow-up expansion. The mechanism modeling of thermal pipe network plays an important role in the simulation and prediction of thermal dynamic characteristics [10,11].

In order to further analyze the characteristics of thermal energy transmission, some scholars have introduced the idea of thermoelectric analogy in the research of heat energy transmission, and achieved interdisciplinary theoretical application through this analogy. Chen et al. [12,13,14,15,16] put forward entransy theory to optimize the heat transfer process by analogy analysis of thermal and electric conduction process, but they mainly focused on the static heat transfer of heat exchangers and ignored the dynamic heat transfer process in the heat network. Feng, Chen and Wang et al. [17,18,19] developed the entransy dissipation extremum principle (EDEP) applying in structural optimizations, which is different from the conventional optimization objectives. They found that the mean temperature differences of the heat conduction assemblies are not always decreased when their internal complexity increases. Lan et al. [20] proposed a method of enthalpy transfer through the equivalent formula relationship between heat supply pipeline and power line, and applied it to the modeling of nodes, pipelines and other components of Multi-carrier Energy System (MCES). Chen et al. [21] introduced the concept of heat charge by referring to the concept of electric charge through the analogy between thermodynamics and electricity theory. They explain that, actually, the heat charge equals to entropy in value, which represents the amount of charging heat at certain temperature. This essentially reveals the similarities and commonalities between thermodynamics and electricity from the perspective of matter, i.e., by mapping the heat charge to the charge. Zhang, Wu and Yang et al. [22,23,24] compared the thermal circuit with temperature as the intensity property and heat flow as the extension property, and obtained the time domain thermal circuit models with distributed parameters and centralized parameters, respectively. This study takes the thermal flow as the extension property for modeling, and introduces the method for frequency domain calculation, which is relatively efficient, but the product of intensity and extension properties does not have the meaning of energy and does not meet the energy postulate. To make up for these shortcomings, Chen et al. [25,26] took temperature and entropy as the intensity and extension properties, respectively. They aimed to achieve the unification of different energy forms, and generalized the Kirchhoff’s law of electricity to suit multi-energy network, but their analysis ignored the quality difference among various energy forms.

In summary, previous literatures mainly focused on the change of energy quantity from multiple scales, but ignored the change of energy quality during heat transportation, especially for dynamic transmission processes. In order to evaluate the quality grade accompanying the quantity variation of thermal transport, this paper proposed the thermal energy scale based on entropy to analyze the transmission process of thermal energy from the perspective of quantity and quality. Firstly, based on the heat transfer law and thermodynamic law, the entropy conduction and convection transport formulas are derived, and the distributed parameter circuit equivalent models of entropy conduction and convection are established. Then, according to the entropy theory and the mathematical equations describing thermal dynamic transmission, the lumped parameter model based on generalized entropy flow is developed. Finally, using the conservation law of energy and the principle of entropy generation, the change of quantity and quality of thermal energy in the transmission process of heat network is analyzed with entropy as the scale, and the correctness of proposed models are proved by numerical simulation and verification procedure with Matlab/simulink.

2. Distribution Parameter Model of Thermal Transmission

2.1. Concept of Entropy Flow

There are three basic modes of thermal energy transfer, namely, heat conduction, heat convection and heat radiation. Heat conduction is a way of heat transfer from high-temperature heat source to low-temperature heat source driven by temperature difference. Thermal convection refers to the heat transfer caused by the relative displacement between various parts of the fluid caused by the macro movement of air, water and other fluids, and the mixing of cold and hot fluids. Thermal radiation refers to the exchange of heat energy between objects in the form of radiant energy emitted by electromagnetic waves due to heat. This paper mainly focus on the heat supply transmission process of direct buried heating pipelines, which are often used for actual heat supply pipe distribution, and the radiation heat transfer between heat transfer fluid (HTF) and the environment can be ignored [27].

Thermal energy transfer and electric energy transfer have similar characteristics. For both, temperature and voltage are considered as their intensity properties, respectively. The current in the circuit is generated by the voltage difference, and the heat conduction in the process of heat conduction is driven by the temperature difference. Here, entropy is introduced to study the dynamic characteristics of heat transfer process. Entropy and entropy flow are compared with the concepts of electric charge and current, and the transmission process of thermal energy is studied in the framework of temperature and entropy flow. Like the product of voltage and current, the product of temperature and entropy has the meaning of energy, which represents the quantity of heat, and meets the energy postulate. The concept of entropy flow in thermodynamics is introduced as shown in Equation (1).

$$\dot{S}=\frac{\Phi }{T_{\mathrm{r}}}$$

(1)

where $\dot{S}$ is the entropy flow in unit of W/K, Φ is the heat flow rate of heat exchange between the system and the external surrounding and T_r is the temperature of heat source. Similar to the product of voltage and current, the product of temperature and entropy flow is also power, and the unit is W.

2.2. Conductive Transport of Entropy

According to Fourier’s heat conduction law, the heat flux in the process of heat conduction is expressed as follows:

$$\Phi =-\lambda A_c\frac{\mathrm{d}T}{\mathrm{d}x}$$

(2)

where λ is the thermal conductivity and A_c is the heat conduction area.

The entropy flow through the object depends on the temperature difference of the whole object. As with heat conduction, for a given temperature difference, the entropy flow is directly proportional to the interface area of the conductor and inversely proportional to the thickness of the conductor, as shown in Equation (3):

$$\{\begin{array}{c}\dot{S}=-k_s{A}_{\mathrm{c}}\frac{\mathrm{d}T}{\mathrm{d}x}\\ k_s=\frac{\lambda }{T}\end{array}$$

(3)

where the negative sign represents that the entropy flow is in the opposite direction of temperature rise and $k_s$ is the entropy conductivity coefficient, which quantifies the entropy conductivity of the material, and the unit is W/m K².

The schematic diagram of entropy conduction is shown in Figure 1. The thickness of the conductor is dx, the temperature on both sides is $T_1$ and $T_2$, respectively, the entropy flow in and out is ${\dot{S}}_1$ and ${\dot{S}}_2$, respectively, and its thermal resistance is $R_{\mathrm{t}}$.

The entropy production [25] can be obtained from the inflow and outflow entropy flow, as shown in Equation (4):

$$\Delta \dot{S}={\dot{S}}_2-{\dot{S}}_1=\frac{T_1-T_2}{R_{\mathrm{t}}{T}_2}-\frac{T_1-T_2}{R_{\mathrm{t}}{T}_1}$$

(4)

where the difference between ${\dot{S}}_1$ and ${\dot{S}}_2$ represents the entropy increase in the heat conduction process. Unlike the current and charge in the circuit, entropy is not conserved. Generally, entropy flows from the high temperature side to the low temperature side, that is to say, it is driven by the temperature difference. However, what leads to the increase of entropy is the irreversibility of the transfer process, while the temperature difference is only one of the influencing factors.

It can be seen that in the process of heat conduction, the entropy generation $\Delta \dot{S}$ is always positive, and only when $T_1=T_2$, $\Delta \dot{S}$ equals to 0. Therefore, it can be concluded that the entropy always increases in the process of heat transfer, where there must be temperature difference, which meets the second law of thermodynamics.

The circuit equivalent model of entropy conduction is shown in Figure 2, in which ${\dot{S}}_{\mathrm{g}}$ is the entropy source marked as the entropy flow rate, indicating that the irreversibility of heat transfer leads to the increase of entropy flow, and $R_s$ is the entropy resistance.

$$R_s=R_{\mathrm{t}}{T}_1$$

(5)

It can be seen that the magnitude of entropy resistance is affected by temperature, so it is not a constant.

In Figure 2, the expression of heat source $T_s$ and the entropy flow rate ${\dot{S}}_{\mathrm{g}}$ are calculated by Equations (6) and (7), respectively.

$$T_s=\frac{{\left(T_1-T_2\right)}^2}{T_2}+T_2$$

(6)

$${\dot{S}}_{\mathrm{g}}=\frac{1}{T_2}\left(T_1-T_2\right){\dot{S}}_1$$

(7)

2.3. Convective Transport of Entropy

Other than conductive transport, entropy is also usually transmitted through fluid such as air or water. The process that entropy is carried by flowing medium is the convective transport of entropy. For simplicity, take the most common water flowing in pipes as an example to illustrate the entropy flow under convection.

As shown in Equation (1), the traditional thermodynamic entropy flow is the ratio of heat flow rate to heat source temperature, which usually describes the energy balance of the thermodynamic system in the process of heat transfer.

However, as a carrier carrying entropy, fluid is not driven by temperature difference. For better description of the role of carrier flow in energy transfer, this paper proposes the concept of “generalized entropy flow”, which is defined as the product of mass flow and specific entropy, as shown in Equation (8). The mass flow reflects flow rate, and the specific entropy reflects heat energy. Therefore, this concept can better describe the flow of energy (work). For convenience of expression, the generalized entropy flow is collectively referred to “entropy flow” in the following text.

$$\dot{S}=\dot{m}s$$

(8)

where $\dot{m}$ is the mass flow in unit of kg/s and s is the specific entropy in unit of J/kg K.

Usually, the heating system uses liquid water as the working medium for heat supply. For incompressible fluid such as water, there is no direct coupling relationship between pressure and temperature. In other words, pressure and temperature can change independently. Since the transport of mass depends on the pressure difference and the specific entropy depends on the temperature, the entropy flow is jointly influenced by the pressure and temperature. Changing the pressure distribution in the pipe and the temperature of the inlet working medium will lead to the change of entropy flow, which corresponds to the concepts of quantity regulation and quality regulation of the heating system, respectively.

Combined with the previous entropy conduction theory, the distributed parameter model of heating pipeline will be established in the following.

In order to facilitate the analysis of the flow and heat transfer processes in the pipe, the pipe is discretized into several continuous pipe finite elements with equal length of $\Delta l$ along the axial direction. The schematic diagram of pipeline discretization is shown in Figure 3. It is approximately considered that the corresponding parameters such as temperature and pressure in the discrete finite elements are uniformly constant in the steady state.

As shown in Figure 4, taking one of the finite element pipe sections for observation, the hot fluid flows in from the left side and out from the right side. Because the pipe wall is not adiabatic, part of the entropy flow flows into the environment through the pipe wall due to convection and conduction, marked as ${\dot{S}}_{\mathrm{con},\mathrm{e}}$. The thermal fluid flows in the pipe, and the mass flow is $\dot{m}$, the entropy flow at the inlet of the pipe is $\dot{m}{s}_{\mathrm{cov},\mathrm{i}-1}$ and the entropy flow out of the pipe is $\dot{m}{s}_{\mathrm{cov},\mathrm{i}}$. $\mathrm{d}{S}_{\mathrm{cv}}/\mathrm{d}t$ refers to the change of entropy with time in the whole control volume. At the inlet of the pipe, the fluid has temperature $T_{i-1}$ and pressure $p_{i-1}$. When it leaves the pipe, the temperature and pressure are $T_i$ and $p_i$, respectively, and $T_e$ is the ambient temperature.

2.4. Steady Entropy Transport

When the system is in a steady state and the inlet flow rate and temperature are constant, the balance equation of energy and entropy can be described as:

$$\dot{m}\left(s_{\mathrm{cov},\mathrm{i}-1}-s_{\mathrm{cov},\mathrm{i}}\right)-{\dot{S}}_{\mathrm{con},\mathrm{e}}+{\dot{S}}_{\mathrm{g}}=0$$

(9)

In the equation above, the first term on the right is the change of inlet and outlet entropy flow, the second term on the right is the entropy flow through the pipe wall and the third term on the right is the entropy production caused by irreversible factors. As mentioned above, water is an incompressible fluid, and its specific entropy mostly depends on the temperature. For a given temperature, the specific entropy of water is calculated as:

$$s=s_{\mathrm{ref}}+c\ln \left(T/T_{\mathrm{ref}}\right)$$

(10)

where c is the specific heat capacity and $s_{\mathrm{ref}}$ and $T_{\mathrm{ref}}$ are the specific entropy and temperature of the reference point, respectively. It is generally specified that the specific entropy starts to be calculated at 0 °C. If zero Celsius degrees is taken as the reference point, then $s_{\mathrm{ref}}$ is 0.

Entropy production rate ${\dot{S}}_{\mathrm{g}}$ is the entropy increase caused by irreversible energy dissipation in the actual thermal process. The calculation formula is as follows [28,29].

$${\dot{S}}_g=\dot{m}\frac{\Delta p}{\rho T_{i-1}}$$

(11)

where $\Delta p$ is the pressure drop in the pipe section due to the frictional resistance of pipeline, which can be calculated by the following formula [20].

$$\Delta p=\frac{8f\Delta l}{d^5\pi {\rho }^2}{\dot{m}}^2$$

(12)

where $\rho$ is the density of working medium, d is the inner diameter of the pipe and f is the energy loss coefficient along the route, and its value is determined by the fluid flow state in the pipe and the roughness of the pipe wall.

From Equations of (8)–(12), equivalent mathematical transformation can be expressed in Equation (13):

$${\dot{m}}^3\frac{8f\Delta l}{d^5\pi {\rho }^3{T}_{i-1}}=\dot{m}\left(c\ln \frac{T_i}{T_{\mathrm{ref}}}-c\ln \frac{T_{i-1}}{T_{\mathrm{ref}}}\right)+\frac{T_{i-1}-T_{\mathrm{e}}}{R_{\mathrm{t}}{T}_{i-1}}$$

(13)

Take the exponent of Equation (13) to obtain the outlet temperature of the control volume under steady-state conditions.

$$T_i=T_{i-1}{e}^{\frac{1}{\dot{m}c}\left({\dot{m}}^3\frac{8f\Delta l}{d^5\pi {\rho }^3{T}_{i-1}}-\frac{T_{i-1}-T_{\mathrm{e}}}{R_{\mathrm{t}}{T}_{i-1}}\right)}$$

(14)

In the steady state, the convective equivalent model of entropy transmission circuit is shown in Figure 5. $R_{\mathrm{s},\mathrm{loss}}$ is the loss entropy resistance and R_s is the branch entropy resistance which is:

$$R_s=R_{\mathrm{t}}{T}_{i-1}$$

(15)

As the HTF exchanges heat with the environment through the pipe wall, part of the entropy flow flows to the surroundings along the branches, which results in decreasing of the total entropy flow. Consequently, the temperature decreases along the primary flow direction.

The loss entropy resistance is calculated as:

$$R_{\mathrm{s},\mathrm{loss}}=\frac{T_{i-1}-T_i}{{\dot{S}}_{i-1}-\frac{T_{i-1}-T_{\mathrm{e}}}{R_{\mathrm{t}}{T}_{i-1}}}$$

(16)

2.5. Unsteady Entropy Transport

Heating system usually adopts the operation mode of quality regulation in most of countries over the world including China, Russia and northern Europe [27,30]. This operation adjusts the water supply temperature of the network at the heat source, which only changes its thermal parameters of HTF, keeping the hydraulic conditions unchanged. This has the advantages of simple operation and stable performance. Different from this, under the mode of mass regulation, the mass flow of water in each branch maintains as a constant, which takes the advantages of precise control. This paper mainly discusses the unsteady equilibrium under the conditions of mass regulation.

The unsteady equation with time term is described as:

$$\frac{\mathrm{d}{S}_{\mathrm{cv}}}{\mathrm{d}t}=\dot{m}\left(s_{\mathrm{cov},\mathrm{i}-1}-s_{\mathrm{cov},\mathrm{i}}\right)-{\dot{S}}_{\mathrm{con},\mathrm{e}}+{\dot{S}}_{\mathrm{g}}$$

(17)

Here the specific entropy capacity $c_s$ is introduced to describe the relationship between temperature and specific entropy, which is defined as:

$$\{\begin{array}{c}ds=c_{s}dT\\ c_s=\frac{c}{T}\end{array}$$

(18)

Entropy capacity is the product of mass and specific entropy capacity, where the mass is taken as the mass of water in the finite element pipe section.

$$C_s=m{c}_s$$

(19)

Thus, Equation (17) can be rewritten as:

$$C_s\frac{\mathrm{d}{T}_{\mathrm{i}-1}}{\mathrm{d}t}=\dot{m}c\left(\ln T_{i-1}-\ln T_i\right)-\frac{T_{i-1}-T_{\mathrm{e}}}{R_s}+{\dot{m}}^3\frac{8f\Delta l}{d^5\pi {\rho }^3{T}_{i-1}}$$

(20)

Equation (20) is the dynamic balance equation of the simplified control volume. The equivalent model of entropy convection transmission is shown in Figure 6. Here, the calculation of loss entropy resistance is the same as Equation (16).

Take the exponent of (20) to obtain he outlet temperature of the control volume in the case of unsteady state.

$$T_i=T_{i-1}{e}^{\frac{1}{\dot{m}c}\left({\dot{m}}^3\frac{8f\Delta l}{d^5\pi {\rho }^3{T}_{i-1}}-\frac{T_{i-1}-T_{\mathrm{e}}}{R_s}-C_s\frac{d{T}_{i-1}}{dt}\right)}$$

(21)

Compared with Equation (14), unstable terms have been involved in Equation (21), which means to take account of the functional expression of outlet temperature when the inlet temperature changes. Thus, if the inlet temperature changes $d{T}_{i-1}=$ 0, it is stable, and the unstable term will be zero, so Equation (21) is the same as Equation (14). This fully demonstrates the correctness of current model derivation.

3. Lumped Parameter Model of Thermal Transmission

The distributed parameter model has high accuracy in calculation, but the equations are relatively complicated, which is not convenient for numerical modeling and analysis of large-scale pipe network. The lumped parameter model has the advantages of simple model and easy expansion. In practical analysis, generally the changes of HTF parameters at the inlet and outlet of pipeline should be paid more attention. Therefore, it is necessary to further express the pipeline with a lumped parameter model. For the finite element pipe section shown in Figure 4, the Partial Differential Equation (PDE) describing the flow of heating working medium can be obtained according to the energy equation for the thermodynamic open system [7,8,9]:

$$c\rho A\frac{\partial T}{\partial t}+c\dot{m}\frac{\partial T}{\partial x}-{\gamma }_0\frac{{\partial }^{2}T}{\partial x^2}+k\left(T-T_{\mathrm{e}}\right)=0$$

(22)

where T is the function of x and t, which means the temperature of the working medium at position x at time t, ${\gamma }_0$ is the thermal diffusion coefficient, $T_{\mathrm{e}}$ is the ambient temperature and is assumed to be constant, A is the cross-sectional area of the pipe and k is the heat loss coefficient. Four items on the left side of the equation are unsteady term, convection term, heat diffusion term and heat loss term, respectively.

Since the effect of the thermal diffusion in central heating system is relative weak, the corresponding item can be ignored in the calculation. Consequently, Equation (22) is simplified as:

$$c\rho A\frac{\partial T}{\partial t}+c\dot{m}\frac{\partial T}{\partial x}+k\left(T-T_{\mathrm{e}}\right)=0$$

(23)

Take the reference temperature as 0 °C, that is, t = 273.15 K and S = 0, then the specific entropy is:

$$s=c\ln \frac{T}{273.15}$$

(24)

Therefore, the specific entropy at ambient temperature is:

$$s_{\mathrm{e}}=c\ln \frac{T_{\mathrm{e}}}{273.15}$$

(25)

Regardless of the reverse Carnot cycle, the lowest temperature of the working fluid is the ambient temperature in actual, and the heat available in the working fluid is the part over ambient temperature. Therefore, this paper focuses on the entropy flow above the ambient temperature.

$$\dot{S}=\dot{m}\left(s-s_{\mathrm{e}}\right)=\dot{m}c\ln \frac{T}{T_{\mathrm{e}}}$$

(26)

From Equations (23) and (26), one dimensional equations with two variables of T and $\dot{S}$ are obtained:

$$\{\begin{array}{c}\frac{\partial T}{\partial x}=-\frac{k{T}_{\mathrm{e}}}{c\dot{m}}\left(e^{\frac{\dot{S}}{\dot{m}c}}-1\right)-\frac{\rho A{T}_{\mathrm{e}}}{\dot{m}}\frac{\partial e^{\frac{\dot{S}}{\dot{m}c}}}{\partial t}\\ \frac{\partial e^{\frac{\dot{S}}{\dot{m}c}}}{\partial x}=-\frac{k}{c\dot{m}{T}_{\mathrm{e}}}\left(T-T_e\right)-\frac{\rho A}{\dot{m}{T}_{\mathrm{e}}}\frac{\partial T}{\partial t}\end{array}$$

(27)

Take $S_a$ as an intermediate variable of the generalized entropy flow $\dot{S}$. Their relationships are as follows:

$$\{\begin{array}{c}{T}_{\mathrm{a}}=T-T_{\mathrm{e}}\\ S_a=e^{\frac{\dot{S}}{c\dot{m}}}-1\\ S_{\mathrm{a}}=\frac{T_{\mathrm{a}}}{T_{\mathrm{e}}}\end{array}$$

(28)

Then Equation (27) can be rewritten as:

$$\{\begin{array}{c}\frac{\partial T_{\mathrm{a}}}{\partial x}=-{\alpha }_1{S}_{\mathrm{a}}-{\beta }_1\frac{\partial S_{\mathrm{a}}}{\partial t}\\ \frac{\partial S_{\mathrm{a}}}{\partial x}=-{\alpha }_2{T}_{\mathrm{a}}-{\beta }_2\frac{\partial T_{\mathrm{a}}}{\partial t}\end{array}$$

(29)

where

$$\{\begin{array}{c}{\alpha }_1=\frac{k{T}_{\mathrm{e}}}{c\dot{m}},{\beta }_1=\frac{\rho A{T}_{\mathrm{e}}}{\dot{m}}\\ {\alpha }_2=\frac{k}{c\dot{m}{T}_{\mathrm{e}}},{\beta }_2=\frac{\rho A}{\dot{m}{T}_{\mathrm{e}}}\end{array}$$

(30)

These partial differential equations include both time and space differential terms, making it difficult to solve the equations directly. In the analysis of classical linear dynamic circuits in complex frequency domain, the method of integral transformation is used to transform the known time domain functions into frequency domain functions, thus transforming the time domain differential equation into algebraic equations in frequency domain. After finding the frequency domain function, the inverse transform will be used to return to the time domain to obtain the differential equation solution that satisfies the initial conditions of the circuit. In this paper, an important integral transformation method, Laplace transform, is used to eliminate the time term of the above-mentioned partial differential equations, which simplifies the equation solving process and facilitates the following analysis.

$$\{\begin{array}{c}{T}_{\mathrm{a}}\left(f,t\right)=L\left[T_{\mathrm{a}}\left(x,t\right)\right]\\ S_{\mathrm{a}}\left(f,t\right)=L\left[S_{\mathrm{a}}\left(x,t\right)\right]\end{array}$$

(31)

where the symbol $L(\cdot )$ is the Laplace operator and f represents the complex frequency domain instead of s that already refers to specific entropy in this paper. The Equation (29) can be expressed as follows in the Laplace domain.

$$\frac{\mathrm{d}{T}_{\mathrm{a}}\left(x,f\right)}{\mathrm{d}x}=-S_{\mathrm{a}}\left(x,f\right)\left({\alpha }_1+f{\beta }_1\right)$$

(32)

$$\frac{\mathrm{d}{S}_{\mathrm{a}}\left(x,f\right)}{\mathrm{d}x}=-T_{\mathrm{a}}\left(x,f\right)\left({\alpha }_2+f{\beta }_2\right)$$

(33)

The Equations (32) and (33) are transformed into second-order linear differential equation and their general solutions are obtained. Then the special solution is determined by the boundary conditions [31]. The results of the solution are:

$$T_{\mathrm{a}2}\left(f\right)=T_{\mathrm{a}1}\left(f\right)\exp \left(-\frac{kl}{c\dot{m}}\right)\exp \left(-f\frac{\rho Al}{\dot{m}}\right)$$

(34)

$$S_{\mathrm{a}2}\left(f\right)=S_{\mathrm{a}1}\left(f\right)\exp \left(-\frac{kl}{c\dot{m}}\right)\exp \left(-f\frac{\rho Al}{\dot{m}}\right)$$

(35)

where $T_{\mathrm{a}1}$ and $S_{\mathrm{a}1}$ are the inlet parameters, $T_{\mathrm{a}2}$ and $S_{\mathrm{a}2}$ are the outlet parameters and l represents the pipe length. The time-domain lumped parameter equation of the heat circuit can be obtained by applying the reverse transformation of Pierre-Simon Laplace to Equations (34) and (35):

$$T_{\mathrm{a}2}\left(t\right)=T_{\mathrm{a}1}\left(t-\frac{\rho Al}{\dot{m}}\right)\exp \left(-\frac{kl}{c\dot{m}}\right)$$

(36)

$$S_{\mathrm{a}2}\left(t\right)=S_{\mathrm{a}1}\left(t-\frac{\rho A}{\dot{m}}l\right)\exp \left(-\frac{kl}{c\dot{m}}\right)$$

(37)

Then bring the relationship between $T_{\mathrm{a}}$ and $S_{\mathrm{a}}$ in Equation (28) to Equations (36) and (37), the following expression can be obtained:

$$T_2\left(t\right)=\left(T_1\left(t-\frac{\rho Al}{\dot{m}}\right)-T_{\mathrm{e}}\right)\exp \left(-\frac{kl}{c\dot{m}}\right)+T_{\mathrm{e}}$$

(38)

$${\dot{S}}_2\left(t\right)=c\dot{m}\ln \left[\left(\exp \left(\frac{{\dot{S}}_1\left(t-\frac{\rho A}{\dot{m}}l\right)}{c\dot{m}}\right)-1\right)\exp \left(-\frac{kl}{c\dot{m}}\right)+1\right]$$

(39)

The Equations (38) and (39) show the functional relationship between the variation of the outlet parameters $T_2$, ${\dot{S}}_2$ and the inlet parameters $T_1$, ${\dot{S}}_1$, which essentially gives the approach to characterize the thermal transmission processes by temperature and generalized entropy flow.

4. Verification and Discussion

In this paper, the equivalent distributed parameter model and lumped parameter model for thermal transport in heating pipeline are established with temperature as intensity property and generalized entropy flow as extension property. In this section, physical and simulation models for thermal transmission based on the proposed models are established by Matlab/simulink. Furthermore, the numerical simulation is conducted under various conditions, dynamic characteristics of which are analyzed and discussed.

4.1. Numerical Simulation and Verification of Distributed Parameter Model

The temperature response of HTF in the trunk and branch of a heat supply network is mainly affected by system parameters such as the flow rate, inlet temperature, pipe length and material. This paper mainly takes the thermal transport process in directly buried pipelines as the research object, so the resistance of radiation heat transfer with the environment can be ignored, and three important aspects need to be considered: (a) the convection heat resistance between HTF and pipelines, (b) the conduction heat resistance of each layer and (c) the soil heat resistance.

For a certain length of heating pipe, the equivalent models of distributed parameter circuits of some pipe sections are obtained by discretizing them. The whole pipe model is obtained by connecting the distributed parameter models. Different degree of pipe discretization will lead to different error temperature response. The influence of different number of nodes on the inlet temperature response is analyzed and verified numerically.

Take l = 100 m of pipe as simulation target and the relevant parameters are shown in

Table 1. Thermodynamic parameters of the heating pipe targeted.

Pipe Length/m	Pipe Diameters/mm	Heat Loss Coefficient/W/(m K)	Thermal Diffusivity/(m2/s)	Mass Flow /(kg/s)
100	DN150	0.420	1.428 × 10−7	3

and

Table 2. Distributed model parameters.

Number of Nodes	Length of Pipe Section /(m)	Ambient Temperature /(K)	Working Medium Specific Heat Capacity/J/(kg K)	Mass of Working Medium in Pipe Section/(kg)	Thermal Resistance /(K/W)
n = 1	100	278.15	4200	1743.66	0.0241
n = 2	50	278.15	4200	871.83	0.0481
n = 5	20	278.15	4200	348.73	0.1202
n = 10	10	278.15	4200	174.37	0.2403
n = 20	5	278.15	4200	87.18	0.4807
n = 50	2	278.15	4200	34.87	1.2017
n = 100	1	278.15	4200	17.44	2.4034

. The inlet temperature of the pipeline at the initial time and the temperature of the water initially stored in the pipeline are both 333.15 K (60 °C), the ambient temperature is 278.15 K (5 °C). The inlet temperature T_in changes with a function of time as shown in Equation (40) after 60 min from the start of operation. Plus, the inlet temperature increases from 333.15 K to 363.15 K (90 °C) with time.

$$T_{\mathrm{in}}\left(t\right)=50\left[\exp \left(t\right)/\left(\exp \left(t\right)+1\right)+0.5\right]+278.15$$

(40)

In Figure 7 and Figure 8, the distributed parameter model is used to simulate the response of outlet temperature and entropy flow under different discrete conditions. The simulation results are compared with solution of the original partial differential equations as expressed in Equation (22).

The red curve in the figures is the solution result of PDE. It can be seen that when the number of nodes n = 1, the outlet temperature responds instantaneously while the inlet temperature changes, and the temperature transmission delay has a large error, which decreases gradually with the increase of the number of nodes. When n = 100, the pipeline is divided into 100 sections, and the temperature response curve is very close to the calculation result of PDE. However, the increase in the number of nodes will lead to an increase in the amount for calculation. In the actual calculation, it is important to choose the appropriate discrete model to meet the requirements.

Figure 9 shows the calculated relative error. When the number of nodes n = 1, the maximum relative error of the outlet temperature is 4.112% at 70 min, and the relative error is reduced to −0.151% at 110 min, that is, 50 min after the inlet temperature starts to change. When n = 100, the maximum relative error is 1.275% at 70 min. Compared with the situation of node number n = 1, the relative error is significantly reduced when n = 100 and the response of the relative error returning to 0 is faster.

4.2. Numerical Simulation and Verification of Lumped Parameter Model

The relevant thermodynamic parameters of the pipeline in Table 1 are also used for numerical simulation and verification for lumped parameter model. Plus, the relationships in Equations (38) and (39) need to be verified. Inlet temperature of the pipe varies as a function of time as shown in Equation (40), with an initial value of 333.15 K. The lumped parameter results are verified by the original PDE. The calculation results for outlet entropy flow and relative error are shown in Figure 10 and Figure 11, where S1 and S2 represent the entropy flow at the pipe inlet and pipe outlet, respectively.

Figure 10 shows the comparison results among the inlet entropy flow, the lumped parameter model and the outlet entropy flow calculated by conventional PDE. When the inlet temperature changes at 0 min, the outlet temperature starts to change after 10.8 min due to the temperature delay. It can be seen that the lumped parameter model nearly has the same time delay and response curve as the PDE curve. The response curve of the model has a certain error compared with PDE, which has been quantitatively analyzed in Figure 11. The maximum relative error is −1.762% at the response point of 10.8 min, and the relative error is −0.001% at the stable point of 30 min. The overall relative error is in the order of 10⁻³, which can be ignored in practical applications, consequently the correctness and effectiveness of the model proposed in this paper are verified.

5. Conclusions

There is obvious distinction in quantity and quality of thermal energy, which is different from electrical energy. To consider the influence of this on thermal energy transport, this paper proposed a generalized entropy flow model and established the distributed and lumped parameter theoretical models for thermal transmission. The main conclusions are drawn as follows:

(1)

The concept of entropy flow was proposed, and a generalized entropy flow model was suggested based on thermodynamic law and the principle of entropy generation. The expression of entropy conduction and convection were deduced, which is the basis of establishing the thermodynamic transmission model based on entropy analysis.

(2)

The corresponding equivalent distributed parameter model was established by introducing temperature and entropy as the intensity and extension properties. It is verified that the models can accurately describe the changes of thermodynamic transmission in the heating network.

(3)

Compared with the partial differential equation describing the flow of heat supply, the lumped parameter model of thermal transport process was derived with generalized entropy flow as scale, which provides a simplified model for the calculation of heating network. The correctness and effectiveness of the proposed model has been verified by Matlab/simulink.

(4)

The theoretical derivation in this paper is extensible, which is expected to be applied for the dynamic characteristics analysis of heating systems using compressible HTF, such as high temperature steam, that needs to take into account the pressure variation for calculation of entropy flow. If so, it is of great benefit to the analysis of thermal energy transmission in coupling with IES and the optimization of multi energy system.