The fire tube boiler is divided into five interconnected subsystems. These subsystems are:

a) Main Fire Tube;

b) First Smoke Chamber;

c) First Smoke Tubes;

d) Second Smoke Chamber;

e) Second Smoke Tubes.

Thermal analysis was performed on each subsystem by applying the first law of thermodynamics [5] and heat transfer correlations [6] . The main objective of thermal analysis is to determine the rate of heat transfer, temperature and flow resistance for each subsystem.

a) Main Fire Tube:

The heat is transferred from the main fire tube to the boiling water mainly by radiation from the formed illuminated flame to the inner surface of the main fire tube. Also, convection from the generated hot combustion gases contributes to the heat transfer to the main fire tube. The thermal analysis of the main fire tube is based on the following assumptions:

1) The combustion process occurs in the main fire tube and is governed by chemical equilibrium [7] .

2) The flame and combustion gases are assumed to be in bulk status with uniform temperature and thermophysical properties.

3) The temperature of inner surface of the main fire tube is mathematically negligible compared with the gas temperature in the radiation formula.

${\stackrel{\dot{}}{Q}}_{input}={\stackrel{\dot{}}{m}}_{f}CV{\eta }_{cc}$(1)

${\stackrel{\dot{}}{m}}_g=\left(1+AF\right){\stackrel{\dot{}}{m}}_f$(2)

${\epsilon }_{wf}=\frac{1}{\frac{1}{{\epsilon }_w}+\frac{A_{mft}}{A_{flame}}\left(\frac{1}{{\epsilon }_f}-1\right)}$(3)

${\stackrel{\dot{}}{Q}}_r={\epsilon }_{wf}\sigma A_{mft}{T}_g^4$(4)

$V_{mft}=\frac{{\stackrel{\dot{}}{m}}_g}{{\rho }_g\left(\frac{\pi D_{mft}^2}{4}\right)}$(5)

$R{e}_{mft}=\frac{{\rho }_g{V}_{mft}{D}_{mft}}{{\mu }_g}$(6)

$h_{mft}=0.023\left(\frac{k_g}{D_{mft}}\right)\left(R{e}_{mft}^{0.8}\right)\left(P{r}_g^{0.3}\right)$(7)

${\stackrel{\dot{}}{Q}}_c=A_{mft}{h}_{mft}\left(T_g-T_{sat}\right)$(8)

${\stackrel{\dot{}}{Q}}_{input}={\stackrel{\dot{}}{Q}}_r+{\stackrel{\dot{}}{Q}}_c+{\stackrel{\dot{}}{m}}_{g}C{p}_g\left(T_g-T_{ref}\right)$(9)

$f_{mft}={\left(1.58\ln \left(R{e}_{mft}\right)-3.28\right)}^{-2}$(10)

$\Delta {\stackrel{\dot{}}{P}}_{mft}=\frac{f_{mft}{L}_{mft}{V}_{mft}^2{\rho }_g}{2D_{mft}}$(11)

*Thermophysical properties are determined at $T_g$.

4) For all the following subsystems, the combustion gases generated from the combustion process in the main fire tube flow through the rest of subsystems in frozen flow. The heat is transferred mainly by convection and radiation from the flowing combustion gas to the boiling water through the subsystem’s surfaces.

b) First Smoke Chamber:

${\stackrel{\dot{}}{Q}}_{rcc1}={\epsilon }_{cc1}\sigma A_{cc1}\left({\stackrel{-----}{T_{cc1}}}^4-T_{sat}^4\right)$ (12)

$V_{cc1}=\frac{{\stackrel{\dot{}}{m}}_g}{{\rho }_g\left(\frac{\pi D_{cc1}^2}{4}\right)}$(13)

$R{e}_{cc1}=\frac{{\rho }_g{V}_{cc1}{D}_{cc1}}{{\mu }_g}$ (14)

$h_{cc1}=0.023\left(\frac{k_g}{D_{cc1}}\right)\left(R{e}_{cc1}^{0.8}\right)\left(P{r}_g^{0.3}\right)$(15)

${\stackrel{\dot{}}{Q}}_{ccc1}=A_{cc1}{h}_{cc1}\Delta T_{lm}$ $\Delta T_{lm}=\frac{\left(T_g-T_{sat}\right)-\left(T_{cc1}-T_{sat}\right)}{\ln \left(\frac{T_g-T_{sat}}{T_{cc1}-T_{sat}}\right)}$(16)

${\stackrel{\dot{}}{m}}_{g}C{p}_g\left(T_g-T_{cc1}\right)={\stackrel{\dot{}}{Q}}_{rcc1}+{\stackrel{\dot{}}{Q}}_{ccc1}$(17)

$f_{cc1}={\left(1.58\ln \left(R{e}_{cc1}\right)-3.28\right)}^{-2}$(18)

$\Delta {\stackrel{\dot{}}{P}}_{cc1}=\frac{f_{cc1}{L}_{cc1}{V}_{cc1}^2{\rho }_g}{2D_{cc1}}$(19)

*Thermophysical properties are determined at $\stackrel{-----}{T_{cc1}}$.

c) First Smoke Tubes:

${\stackrel{\dot{}}{Q}}_{r1}={\epsilon }_1\sigma A_1\left({\stackrel{------}{T_{g1}}}^4-T_{sat}^4\right)$(20)

$V_1=\frac{{\stackrel{\dot{}}{m}}_g}{n_{t1}{\rho }_g\left(\frac{\pi D_1^2}{4}\right)}$(21)

$R{e}_1=\frac{{\rho }_g{V}_1{D}_1}{{\mu }_g}$(22)

$h_1=0.023\left(\frac{k_g}{D_1}\right)\left(R{e}_1^{0.8}\right)\left(P{r}_g^{0.3}\right)$(23)

${\stackrel{\dot{}}{Q}}_{c1}=A_1{h}_1\Delta T_{lm}$ $\Delta T_{lm}=\frac{\left(T_{cc1}-T_{sat}\right)-\left(T_{g1}-T_{sat}\right)}{\ln \left(\frac{T_{cc1}-T_{sat}}{T_{g1}-T_{sat}}\right)}$(24)

${\stackrel{\dot{}}{m}}_{g}C{p}_g\left(T_{cc1}-T_{g1}\right)={\stackrel{\dot{}}{Q}}_{r1}+{\stackrel{\dot{}}{Q}}_{c1}$(25)

$f_1={\left(1.58\ln \left(R{e}_1\right)-3.28\right)}^{-2}$(26)

$\Delta {\stackrel{\dot{}}{P}}_1=\frac{f_1{L}_1{V}_1^2{\rho }_g}{2D_1}$(27)

*Thermophysical properties are determined at $\stackrel{------}{T_{g1}}$.

d) Second Smoke Chamber:

${\stackrel{\dot{}}{Q}}_{rcc2}={\epsilon }_{cc2}\sigma A_{cc2}\left({\stackrel{-----}{T_{cc2}}}^4-T_{sat}^4\right)$(28)

$V_{cc2}=\frac{{\stackrel{\dot{}}{m}}_g}{{\rho }_g\left(\frac{\pi D_{cc2}^2}{4}\right)}$(29)

$R{e}_{cc2}=\frac{{\rho }_g{V}_{cc2}{D}_{cc2}}{{\mu }_g}$(30)

$h_{cc2}=0.023\left(\frac{k_g}{D_{cc2}}\right)\left(R{e}_{cc2}^{0.8}\right)\left(P{r}_g^{0.3}\right)$(31)

${\stackrel{\dot{}}{Q}}_{ccc2}=A_{cc2}{h}_{cc2}\Delta T_{lm}$ $\Delta T_{lm}=\frac{\left(T_{g1}-T_{sat}\right)-\left(T_{cc2}-T_{sat}\right)}{\ln \left(\frac{T_{g1}-T_{sat}}{T_{cc2}-T_{sat}}\right)}$(32)

${\stackrel{\dot{}}{m}}_{g}C{p}_g\left(T_{g1}-T_{cc2}\right)={\stackrel{\dot{}}{Q}}_{rcc2}+{\stackrel{\dot{}}{Q}}_{ccc2}$(33)

$f_{cc2}={\left(1.58\ln \left(R{e}_{cc2}\right)-3.28\right)}^{-2}$(34)

$\Delta {\stackrel{\dot{}}{P}}_{cc2}=\frac{f_{cc2}{L}_{cc2}{V}_{cc2}^2{\rho }_g}{2D_{cc2}}$(35)

*Thermophysical properties are determined at $\stackrel{------}{T_{cc2}}$.

e) Second Smoke Tubes:

${\stackrel{\dot{}}{Q}}_{r2}={\epsilon }_2\sigma A_2\left({\stackrel{------}{T_{g2}}}^4-T_{sat}^4\right)$ (36)

$V_2=\frac{{\stackrel{\dot{}}{m}}_g}{n_{t2}{\rho }_g\left(\frac{\pi D_2^2}{4}\right)}$ (37)

$R{e}_2=\frac{{\rho }_g{V}_2{D}_2}{{\mu }_g}$ (38)

$h_2=0.023\left(\frac{k_g}{D_2}\right)\left(R{e}_2^{0.8}\right)\left(P{r}_g^{0.3}\right)$ (39)

${\stackrel{\dot{}}{Q}}_{c2}=A_2{h}_2\Delta T_{lm}$ $\Delta T_{lm}=\frac{\left(T_{cc2}-T_{sat}\right)-\left(T_{g2}-T_{sat}\right)}{\ln \left(\frac{T_{cc2}-T_{sat}}{T_{g2}-T_{sat}}\right)}$ (40)

${\stackrel{\dot{}}{m}}_{g}C{p}_g\left(T_{cc2}-T_{g2}\right)={\stackrel{\dot{}}{Q}}_{r2}+{\stackrel{\dot{}}{Q}}_{c2}$ (41)

$f_2={\left(1.58\ln \left(R{e}_2\right)-3.28\right)}^{-2}$ (42)

$\Delta {\stackrel{\dot{}}{P}}_2=\frac{f_2{L}_2{V}_2^2{\rho }_g}{2D_2}$ (43)

*Thermophysical properties are determined at $\stackrel{------}{T_{g2}}$.

Then the boiler efficiency is determined by:

$\eta =100.0\frac{m_{st}\left(h_{st}-h_w\right)={\displaystyle \sum \text{Heat}\text{Transfer}}}{{\stackrel{\dot{}}{Q}}_{input}}$

The combustion process that takes place in the main fire tube is expressed according to the following chemical reaction formula:

$\begin{array}{l}{\text{C}}_x{\text{H}}_y+z\left({\text{O}}_2+3.76{\text{N}}_{\text{2}}\right)\\ \to n_{\text{O}}\text{O}+n_{\text{H}}\text{H}+n_{\text{OH}}\text{OH}+n_{{\text{H}}_{\text{2}}\text{O}}{\text{H}}_{\text{2}}\text{O}+n_{\text{N}}\text{N}+n_{\text{NO}}\text{NO}+n_{\text{CO}}\text{CO}\\ +n_{{\text{CO}}_2}{\text{CO}}_2+n_{{\text{O}}_2}{\text{O}}_2+n_{{\text{H}}_2}{\text{H}}_2+n_{{\text{N}}_2}{\text{N}}_2\end{array}$

where the combustion products are found based on the chemical equilibrium model, where the number of moles for the combustion product is governed by the following set of equations:

$n_{\text{O}}=K{p}_1{\left(\frac{P_3}{n_t}\right)}^{-0.5}{\left(n_{{\text{O}}_{\text{2}}}\right)}^{0.5}$(44)

$n_{\text{OH}}=K{p}_3{\left(n_{{\text{O}}_{\text{2}}}\right)}^{0.5}{\left(n_{{\text{H}}_{\text{2}}}\right)}^{0.5}$(45)

$n_{\text{N}}=K{p}_5{\left(\frac{P_3}{n_t}\right)}^{-0.5}{\left(n_{{\text{N}}_{\text{2}}}\right)}^{0.5}$(46)

$n_{\text{CO}}=K{p}_9\left(n_{\text{C}}\right){\left(n_{{\text{O}}_{\text{2}}}\right)}^{0.5}$(47)

$N_{\text{H}}=y=n_{\text{H}}+n_{\text{OH}}+2n_{{\text{H}}_{\text{2}}\text{O}}+2n_{{\text{H}}_{\text{2}}}$(48)

$N_{\text{N}}=2\ast 3.76z=n_{\text{N}}+n_{\text{NO}}+2n_{{\text{N}}_{\text{2}}}$(49)

$n_{\text{H}}=K{p}_2{\left(\frac{P_3}{n_t}\right)}^{-0.5}{\left(n_{{\text{H}}_{\text{2}}}\right)}^{0.5}$(50)

$n_{{\text{H}}_{\text{2}}\text{O}}=K{p}_4{\left(\frac{P_3}{n_t}\right)}^{0.5}\left(n_{{\text{H}}_{\text{2}}}\right){\left(n_{{\text{O}}_{\text{2}}}\right)}^{0.5}$(51)

$n_{\text{NO}}=K{p}_6{\left(n_{{\text{N}}_{\text{2}}}\right)}^{0.5}{\left(n_{{\text{O}}_{\text{2}}}\right)}^{0.5}$(52)

$n_{{\text{CO}}_{\text{2}}}=K{p}_{10}\left(n_{{\text{O}}_{\text{2}}}\right)\left(n_{\text{C}}\right)$(53)

$N_{\text{O}}=2z=n_{\text{O}}+n_{\text{OH}}+n_{{\text{H}}_{\text{2}}\text{O}}+n_{\text{NO}}+n_{\text{CO}}+2n_{{\text{CO}}_{\text{2}}}+2n_{{\text{O}}_{\text{2}}}$(54)

$N_{\text{C}}=x=n_{\text{CO}}+n_{{\text{CO}}_{\text{2}}}$(55)

$N_{\text{H}}=y=n_{\text{H}}+n_{\text{OH}}+2n_{{\text{H}}_{\text{2}}\text{O}}+2n_{{\text{H}}_{\text{2}}}$(56)

$N_{\text{N}}=2\ast 3.76z=n_{\text{N}}+n_{\text{NO}}+2n_{{\text{N}}_{\text{2}}}$(57)

*where the equilibrium constants depend on the combustion temperature.

The shell of fire tube boiler is subjected to inner hoop stress, which is calculated as:

$\sigma =\frac{P_{st}\ast D_{shell}}{2x}$ (58)

The mathematical equations with all heat transfer, thermodynamic, and chemical equilibrium data are solved through a visual FORTRAN program.

Table 1 shows the results obtained from the model for the studied case (1 TPH Boiler), while the boiler dimensions and heat transfer areas are:

$A_{mft}=\pi \left(0.55\right)\left(2.1\right)=3.6285{\text{m}}^2$

$\begin{array}{c}{A}_{cc1}=\frac{\pi }{4}\left[{\left(0.99\right)}^2-{\left(0.5\right)}^2+{\left(0.99\right)}^2-{\left(0.55\right)}^2-20{\left(0.051\right)}^2\right]+\pi \left(0.99\right)\left(0.6\right)\\ =2.931{\text{m}}^2\end{array}$

$A_1=\pi \left(20\right)\left(0.051\right)\left(2.15\right)=6.89{\text{m}}^2$

$A_{cc2}=1.2{\text{m}}^2$

$A_2=\pi \left(20\right)\left(0.051\right)\left(2.9\right)=9.293{\text{m}}^2$

Table 2 shows the results obtained from the model for the studied case (2 TPH Boiler), while the boiler dimensions and heat transfer areas are:

$A_{mft}=\pi \left(0.62\right)\left(2.58\right)=5.0253{\text{m}}^2$

$\begin{array}{c}{A}_{cc1}=\frac{\pi }{4}\left[{\left(1.05\right)}^2-{\left(0.5\right)}^2+{\left(1.05\right)}^2-{\left(0.62\right)}^2-36{\left(0.051\right)}^2\right]+\pi \left(1.05\right)\left(0.6\right)\\ =3.74{\text{m}}^2\end{array}$

$A_1=\pi \left(36\right)\left(0.051\right)\left(2.6\right)=15{\text{m}}^{\text{2}}$

$A_{cc2}=1.2{\text{m}}^2$

$A_2=\pi \left(36\right)\left(0.051\right)\left(3.4\right)=19.6{\text{m}}^2$

Table 3 shows the results obtained from the model for the studied case (3 TPH Boiler), while the boiler dimensions and heat transfer areas are:

$A_{mft}=\pi \left(0.72\right)\left(2.75\right)=6.22{\text{m}}^2$

$\begin{array}{c}{A}_{cc1}=\frac{\pi }{4}\left[{\left(1.2\right)}^2-{\left(0.5\right)}^2+{\left(1.2\right)}^2-{\left(0.72\right)}^2-50{\left(0.0483\right)}^2\right]+\pi \left(1.2\right)\left(0.6\right)\\ =3.83{\text{m}}^2\end{array}$

$A_1=\pi \left(50\right)\left(0.0483\right)\left(2.6\right)=19.73{\text{m}}^2$

$A_{cc2}=1.2{\text{m}}^2$

$A_2=\pi \left(50\right)\left(0.0483\right)\left(3.5\right)=25.8{\text{m}}^2$