3.2.1. Burner

(a) *Qar,net* is the lower calorific value of natural gas (NG), according to the value of the Utility Boiler Manual [42]:

$$Q\_{ar,net} = 50200 \text{kJ} \tag{9}$$

(b) *Qair* is the energy of air and the value is approximately zero:

$$Q\_{\dot{a}\dot{r}} = 0\tag{10}$$

(c) Energy loss of incomplete combustion is the product of incomplete combustion coefficient ε and the lower calorific value of natural gas (NG) *Qar,net*. In the calculation, the value of incomplete combustion coefficient ε is 0.3:

$$Q\_{b\,inc} = \varepsilon Q\_{ar\,net} \tag{11}$$

(d) The calculation of energy loss of heat leakage of burner is abstracted as a mathematical model of the heat transfer process of a cylinder tube with gas flowing in air, so are the energy loss of heat leakage of pipe and the extended part of well, as shown in Figure 5. The calculation of heat leakage energy is based on Fourier's Law and Newton's Law of Cooling of heat transfer theory:

$$Q\_{b,l} = \frac{2\pi (t\_{f,b} - t\_0)L\_b}{\frac{2}{h\_{b,1}d\_{b,1}} + \frac{1}{\lambda\_b} \ln \frac{d\_{b,2}}{d\_{b,1}} + \frac{2}{h\_{b,2}d\_{b,2}}} \times \frac{1}{G\_{NG}} \times 10^{-3} \tag{12}$$

$$Q\_{p,l} = \frac{2\pi (t\_{f,p} - t\_0)L\_p}{\frac{2}{h\_{p,1}d\_{p,1}} + \frac{1}{\lambda\_p}\ln\frac{d\_{p,2}}{d\_{p,1}} + \frac{2}{h\_{p,2}d\_{p,2}}} \times \frac{1}{G\_{NG}} \times 10^{-3} \tag{13}$$

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

$$Q\_{w,l} = \frac{2\pi (t\_{f,w} - t\_0) L\_o}{\frac{2}{h\_{w,1} d\_{w,1}} + \frac{1}{\lambda\_w} \ln \frac{d\_{w,2}}{d\_{w,1}} + \frac{2}{h\_{w,2} d\_{w,2}}} \times \frac{1}{G\_{NG}} \times 10^{-3} \tag{14}$$

The influence of air leaks is ignored. The convective heat transfer coefficient is different when there's wind and there's no wind. So the calculation of the convective heat transfer coefficient outside the tube is divided into forced convection and natural convection.

(e) Energy loss of flow *Qb,f* concludes path energy loss and local energy loss, the calculation is based on the algorithm in fluid mechanics:

$$Q\_{b,f} = \lg(\varepsilon\_b \frac{L\_b}{d\_{b,1}} \frac{u\_b^2}{2g} + 2\xi\_b \frac{u\_b^2}{2g}) \times \frac{G\_\varepsilon}{G\_{NG}} \times 10^{-3} \tag{15}$$

The flow velocity *ub* is calculated by mass flow rate and pipe diameter.

The flow in the tube is in the turbulent smooth zone in reality, so coefficient of path energy loss of the burner is determined by Equation (16), the flow in pipe and well as well. Therefore, the calculation equation of the coefficient of local energy loss of pipe and well will not be repeated below:

$$
\omega\_b = \frac{0.3164}{\mathrm{Re}\_b^{0.25}} \tag{16}
$$

A right angle loss and a valve loss are considered for coefficient of local energy loss of the burner. The coefficient of local energy loss of the limit value of a pipe section expansion is 1 and a valve opening of 50% is 1.8. And the value of local loss coefficient is different for different components as show in Figure 1:

$$
\zeta\_b = 1 + 1.8 \tag{17}
$$

(f) Energy out of the burner *Qb*,*to*,*<sup>p</sup>* is solved by the conservation of energy equation. The calculation of the energy out of the pipe, well and soil is similar, so the equation will not be repeated below:

$$Q\_{b,to,p} = Q\_{ar,net} + Q\_{air} - Q\_{b,in} - Q\_{b,l} - Q\_{b,f} \tag{18}$$

3.2.2. Pipe

> Energy loss of flow *Qp*, *f* includes path energy loss only:

$$Q\_{p,f} = g\varepsilon\_p \frac{L\_p}{d\_{p,2}} \frac{u\_p^2}{2g} \times \frac{G\_\varepsilon}{G\_{NG}} \times 10^{-3} \tag{19}$$

3.2.3. Well

> (a) Energy of exhaust gas that flows from the outlet of the well to the environment:

$$Q\_{w,out} = \frac{1}{G\_{NG}} \times G\_c c\_{p,c} t\_{w,out} \tag{20}$$

(b) Energy loss of flow *Qw,f* includes path energy loss and local energy loss:

$$Q\_{w,f} = g(\varepsilon\_{w}\frac{L\_{w}}{d\_{w,1}}\frac{u\_{w}^{2}}{2g} + 2\zeta\_{w}\frac{u\_{w}^{2}}{2g}) \times \frac{G\_{\varepsilon}}{G\_{NG}} \times 10^{-3} \tag{21}$$

Coefficient of local energy loss of the well:

$$
\xi\_w = 2.993 + 0.985 \,\tag{22}
$$
