*3.1. Balance Models*

The balance models are based on energy loss and exergy loss of each component in the process of energy flow and exergy flow. Figure 5 shows the energy loss of each component of the polluted-soil thermal remediation system. At the beginning of the energy flow throughout the system, the natural gas (NG) and air carry energy through their respective pipes into the burner. When the gas flows through the pipeline, there are throttling and friction process in the flow, which cause an energy loss. Throttling is a local flow loss, while friction is a path loss of flow. In reality, as long as there is flow in the pipeline, there will be flow loss, and as long as there is a pipe with fluid exposed to the environment, there will be heat leakage loss.

In the burner, incomplete combustion caused by inadequate mixes of fuel and air or the low temperature in the combustor cause energy losses. There are also heat leakage, air leakage and flow loss in the burner. After the energy loss is removed, the remaining energy flows out of the burner and through the pipe into the heating well. There are heat leakage and flow loss in the pipe. Local flow loss exists in the heating well because of the bent pipe. Part of the energy flowing to the heating well is transferred to the soil, heating it. The remaining energy is discharged directly to the environment by the outlet of the heating well through high-temperature exhaust gas, resulting in the maximal energy loss of the whole system. In addition to heating up the soil, the energy in the soil will also lose heat to the surrounding non-heating soil zone and to the air through the surface insulation layer.

Figure 6 shows the exergy loss of each component of the polluted-soil thermal remediation system. Energy loss is accompanied by exergy loss, so all of the energy loss described above has the consequent loss of exergy, including incomplete combustion, heat leakage, flow leakage and so on. Besides, Irreversible combustion, heat transfer, non-isothermal heat release and non-isothermal heat absorption also cause the exergy loss. Consequently, the energy and exergy balance of each component are modeled as shown in Sections 3.1.1 and 3.1.2, respectively.

**Figure 5.** The locations of energy loss of components of polluted-soil thermal remediation system.

**Figure 6.** The locations of exergy loss of components of polluted-soil thermal remediation system.

## 3.1.1. Energy Balance Models

Based on the balance of energy principle and energy loss of each component described in Figure 5, the energy balance equation is established as follows. Equations (1)–(4) are the energy balance models of the burner, pipe, well and soil separately:

$$Q\_{ar,net} + Q\_{air} = Q\_{b,to,p} + Q\_{b,inc} + Q\_{b,l} + Q\_{b,f} \tag{1}$$

$$Q\_{p,in} = Q\_{b,to,p} = Q\_{p,to,w} + Q\_{p,l} + Q\_{p,f} \tag{2}$$

*Energies* **2019**, *12*, 4018

$$Q\_{w,in} = Q\_{p,to,w} = Q\_{w,to,s} + Q\_{w,out} + Q\_{w,l} + Q\_{w,f} \tag{3}$$

$$Q\_{s,in} = Q\_{w,to,s} = Q\_{s,a} + Q\_{s,l} \tag{4}$$

## 3.1.2. Exergy Balance Models

The exergy balance models is similar to the energy balance model, based on the balance of exergy principle and exergy loss of each component described in Figure 6. Equations (5)–(8) are the exergy balance models of the burner, pipe, well and soil, respectively:

$$E\_r + E\_{air} = E\_{b,to,p} + E\_{b,irr} + E\_{b,inc} + E\_{b,l} + E\_{b,f} \tag{5}$$

$$E\_{p,in} = E\_{b,to,p} = E\_{p,to,np} + E\_{p,l} + E\_{p,f} \tag{6}$$

$$E\_{w,in} = E\_{p,to,w} = E\_{w,to,s} + E\_{x,Q} + E\_{x,Q\_1} + E\_{w,out} + E\_{w,f} + E\_{w,f} \tag{7}$$

$$E\_{\rm s,in} = E\_{\rm w,to,s} = E\_{\rm s,a} + E\_{\rm x,Q\_2} + E\_{\rm s,l} \tag{8}$$
