**Appendix A**




**Table A2.** The meanings of the parameters in flow diagrams.

#### **Appendix B Calculation of Intermediate Parameters**

*(a) Excess Air Coe*ffi*cient and Mass Flow Rates Calculation*

Mass flow rate of exhaust gas is calculated by state of soil, duration of stages and the set temperature of the exhaust. However, in the basic method (BM) the required conditions for calculation in each stage are uniform, and in variable-condition mode (VCM) those are distinguishing because of different exhaust gas temperatures at different stages. Excess air coefficient is also different caused by varied adiabatic combustion temperature.

Thermal requirement is the quantity of heat in the heat transfer process from well to soil by exhaust gas. *Q*I is the heat in the first stage, *Q*II is in the second stage and *Q*III is the third. *cp,s* is specific heat with the moisture content of 0.3:

$$\begin{cases} \begin{array}{c} Q\_{\text{I}} = m\_{s} \mathfrak{c}\_{p,s} (100 - t\_{0}) \\ Q\_{\text{II}} = m\_{\mathfrak{w}} \gamma \end{array} \\ \begin{array}{c} Q\_{\text{III}} = m\_{\mathfrak{p}s} \gamma \\ Q\_{\text{III}} = m\_{\mathfrak{p}s} \mathfrak{c}\_{p,\mathfrak{p}s} (t\_{\mathfrak{s}} - 100) \end{array} \end{cases} \tag{A1}$$

Mass flow rate of exhaust gas in BM of the three stages are calculated uniformly, as follows. And in VCM the mass flow rates of exhaust gas of the three stages are calculated respectively using separate thermal requirement. Because the calculation method is same as that in BM, it will not be repeated here.

Mass flow rate of exhaust gas in BM:

$$\mathbf{G}\_{\ell} = \frac{\mathbf{Q}\_{\ell} + \mathbf{Q}\_{\mathrm{II}} + \mathbf{Q}\_{\mathrm{III}}}{\pi c\_{p,\ell} (t\_{\mathrm{up},\mathrm{in}} - t\_{\mathrm{IV},\mathrm{out}})} = \frac{m\_{\ell} c\_{p,\ell} (100 - t\_0) + m\_{\mathrm{IV}} \gamma + m\_{\mathrm{P}} c\_{p,\mathrm{IV}} (t\_{\mathrm{\ell}} - 100)}{\pi c\_{p,\ell} (t\_{\mathrm{up},\mathrm{in}} - t\_{\mathrm{IV},\mathrm{out}})} \tag{A2}$$

In order to calculate the excess air coefficient α, chemical equation: Equation (48) and energy equation: Equation (49) in burner is essential. The excess air coefficient α is the ratio of redundant air volume to the right air volume in a complete reaction:

$$\text{C}\text{CH}\_4 + 2(1+a)\text{O}\_2 + \frac{2 \times 78}{21}(1+a)\text{N}\_2 \rightarrow \text{CO}\_2 + 2\text{H}\_2\text{O} + \frac{2 \times 78}{21}(1+a)\text{N}\_2 + 2a\text{O}\_2 + Q \tag{A3}$$

$$G\_{\rm NCG}g\_{\rm nF,alt}(1-\varepsilon) + G\_{\rm NCG}c\_{p,\rm NCG}t\_{\rm NCG} + G\_{\rm air}c\_{p,\rm air}t\_{\rm air} = G\_{\rm t}c\_{p,\rm b}t\_{\rm b} \tag{A4}$$

*cp,NG* is the specific heat of natural gas (NG) and *cp,air* is the specific heat of air. Because the temperature of natural gas (NG) and air are close to zero, the two parameters are not involved in the calculation.

Equation (50) is the excess air coefficient α:

$$\alpha = \frac{\left(\frac{16q\_{\nu,\text{at}}(1-\iota)}{c\_{\flat\flat}t\_{\flat}} - 288.012\right)}{272.012} \tag{A5}$$

Mass flow rate of natural gas (NG) is corresponding to mass flow rate of exhaust gas one by one in each stage:

$$G\_{NG} = \frac{G\_{\ell} \times 16}{272.012\alpha + 288.012} \tag{A6}$$

*(b) Convective Heat Transfer Coe*ffi*cient Calculation*

Convective heat transfer coefficient in the calculation of energy loss of heat leakage is divided to three categories:

(a) Convective heat transfer coefficient of forced convection in tube is as follows:

$$\begin{cases} \begin{array}{c} \text{Re} = \frac{\text{\"} \text{\"}}{\text{\text{\"}} \text{\"}}\\ \text{\"} \text{\"} \end{array} \begin{array}{c} \text{\"} \text{\"} \text{\"} \text{\"} = \frac{\text{\"} \text{\"}}{\text{\text{\"}} \text{\"}}\\ \text{\"} \text{\"} \text{\"} \end{array} \begin{array}{c} \text{\"} \text{\"} \text{\"} \text{\"} \end{array} \tag{A}$$

(b) Convective heat transfer coefficient of forced convection outside the tube is solved in Equation (53). The speed of the wind is 3 meters per second in the paper:

$$\begin{cases} \begin{array}{c} \text{Re}\_{\text{altr}} = \frac{\mu\_{\text{alt}}d}{\nu\_{\text{altr}}}\\ \text{Nu}\_{\text{altr}} = \text{C.Re}\_{\text{altr}}^{\text{n}} \text{Pr}^{1/3}\_{\text{altr}}\\ h\_{\text{altr}} = \frac{\lambda\_{\text{altr}}}{d} \text{Nu}\_{\text{altr}} \end{array} \end{cases} \tag{A8}$$

(c) Equation (54) is the convective heat transfer coefficient of natural convection outside the tube:

$$\begin{cases} \begin{array}{c} \text{Gr}\_{\text{alr}} = \frac{\\$^{\Omega}\text{V}\text{-}\!\!\!\text{H}^{\text{L}}}{\text{V}^{2}}\\ \text{Nu}\_{\text{alr}} = \text{C}(\text{Gr}\_{\text{alr}}\text{-}\text{Pr}\_{\text{alr}})^{\text{R}}\_{\text{m}}\\ \end{array} \end{cases} \tag{A9}$$

*(c) Tube Surface Temperature Calculation*

$$\begin{cases} \ t\_{b,\mu,\rho} = \frac{Q\_b t}{L\_\nu b\_{b,2} \pi d\_{b,2}} + t\_0\\ T\_{b,\mu,\rho} = t\_{b,\mu,\rho} + 273.15 \end{cases} \tag{A10}$$

$$\begin{cases} \ t\_{p,\mu,\rho} = \frac{Q\_{p,l}}{L\_p h\_{p,2} \pi d\_{p,2}} + t\_0\\ T\_{p,\mu,\rho} = t\_{p,\mu,\rho} + 273.15 \end{cases} \tag{A11}$$

$$\begin{cases} \ t\_{w,w\rho} = \frac{Q\_{\nu\mathcal{I}}}{L\_{\nu}h\_{\nu 2}\pi d\_{\nu 2}} + t\_0\\ \ T\_{\mathcal{U}\mathcal{U}\mathcal{U}\rho} = t\_{\mathcal{U}\mathcal{U}\rho} + 273.15 \end{cases} \tag{A12}$$

#### *(d) Temperature of Intermediate Heat Source Calculation*

In the actual heat transfer process, the temperature of the heat source will change as Figure A1 shows, so the temperature of intermediate heat source is essential to calculate. The logarithmic mean temperature difference is used to calculate the temperature of intermediate high temperature heat source *TH*, so as the temperature of intermediate low temperature heat source *TL*, and the *TL* is different in each stage:

$$\overline{T\_H} = \frac{T\_{w,iu} - T\_{w,out}}{\ln \frac{T\_{w,iu}}{T\_{w,out}}} \tag{A13}$$

$$\begin{cases} \begin{array}{c} \overline{T\_{L,l}} = \frac{3\Im 3 - T\_0}{\ln \frac{3\Im 3}{T\_0}}\\ \overline{T\_{L,\Pi}} = 3\Im 3\\ \overline{T\_{L,\Pi}} = \frac{T\_s - 3\Im 3}{\ln \frac{3\Im 3}{3\Im 3}} \end{array} \end{cases} \tag{A14}$$

**Figure A1.** The heat transfer process: (**a**) The ideal heat transfer and (**b**) The actual heat transfer.
