*3.3. Smoke Height*

*3.3. Smoke Height*  In fire research, the common methods for solving smoke layer height mainly include: the extinction method, *N* percentage method, minimum integral ratio method and piecewise linear method. Among them, the minimum integral ratio method and the piecewise linear method are used to process the temperature data. In this study, the minimum integral ratio method is used to solve the height of the smoke layer. The method has a relatively clear definition of the neutral plane and a more rigorous calculation and derivation process, and the calculation results are more convincing. It defines the height of the smoke layer as the position where the sum of the minimum integral ratios of the upper and lower In fire research, the common methods for solving smoke layer height mainly include: the extinction method, *N* percentage method, minimum integral ratio method and piecewise linear method. Among them, the minimum integral ratio method and the piecewise linear method are used to process the temperature data. In this study, the minimum integral ratio method is used to solve the height of the smoke layer. The method has a relatively clear definition of the neutral plane and a more rigorous calculation and derivation process, and the calculation results are more convincing. It defines the height of the smoke layer as the position where the sum of the minimum integral ratios of the upper and lower layers takes the minimum value. In the calculation, the integral ratios of the upper hot smoke layer and the lower air layer at time *t* are, respectively:

$$r\_{\rm u} = \frac{1}{\left(H - z\right)^2} \cdot \int\_z^H T(z)dz \cdot \int\_z^H \frac{1}{T(z)}dz\tag{2}$$

$$\sigma\_{l} = \frac{1}{z^{2}} \cdot \int\_{0}^{z} T(z) dz \cdot \int\_{0}^{z} \frac{1}{T(z)} dz \tag{3}$$

௭ ௭ ௭ Then, the height *z* of the smoke layer at time *t* should satisfy:

$$r\_l = \min(r\_u + r\_l) \tag{4}$$

(2)

(3)

Then, the height *z* of the smoke layer at time *t* should satisfy: ௧ = min (௨ + ) (4) Based on the above method and the temperature data of the 1# thermocouple tree, MATLAB is used to solve the height of the smoke layer, and finally the height of the smoke layer under different air supply configurations is given as shown in Figure 10. It can be Based on the above method and the temperature data of the 1# thermocouple tree, MATLAB is used to solve the height of the smoke layer, and finally the height of the smoke layer under different air supply configurations is given as shown in Figure 10. It can be seen from the figure that within the range of air volume and air supply port height in this experiment, with the gradual increase in the forced air supply volume, the height of the smoke layer also gradually increases; with the increase in the air supply inlet height, the height of the smoke layer gradually decreases.

seen from the figure that within the range of air volume and air supply port height in this experiment, with the gradual increase in the forced air supply volume, the height of the smoke layer also gradually increases; with the increase in the air supply inlet height, the

height of the smoke layer gradually decreases.

**Figure 10.** Comparison of cabin smoke layer heights under different supply air volume conditions. **Figure 10.** Comparison of cabin smoke layer heights under different supply air volume conditions. Based on the above data, let the dimensionless smoke layer height <sup>∗</sup> = /, the

Based on the above data, let the dimensionless smoke layer height <sup>∗</sup> = /, the dimensionless air inlet height ℎ<sup>∗</sup> = ℎ/, and introduce the dimensionless parameters *P* and *Q* to make the data dimensionless. The curves of *P* and *Q* after processing are shown in Figure 11. It can be seen that *P* and *Q* are approximately linearly related: = Based on the above data, let the dimensionless smoke layer height *H*<sup>∗</sup> = *H*/*H*0, the dimensionless air inlet height *h* <sup>∗</sup> = *h*/*H*0, and introduce the dimensionless parameters *P* and *Q* to make the data dimensionless. The curves of *P* and *Q* after processing are shown in Figure 11. It can be seen that *P* and *Q* are approximately linearly related: *P* = 1.72 × *Q* + 50.79. dimensionless air inlet height ℎ<sup>∗</sup> = ℎ/, and introduce the dimensionless parameters *P* and *Q* to make the data dimensionless. The curves of *P* and *Q* after processing are shown in Figure 11. It can be seen that *P* and *Q* are approximately linearly related: = 1.72 × + 50.79.

$$P = F(H^\*, N\_\prime h^\*) = H^\*/h^\* \times N \times (1.9h^\* + 0.7) \tag{5}$$

$$Q = G(N, h^\*) = 3.7/h^\* \times \exp(0.028N) \tag{6}$$

1.72 × + 50.79.

air supply volume.

**0 100 200 300 400 500 600 700 800 0 R2 ≈0.98 Figure 11.** The relationship between the dimensionless smoke layer height and the dimensionless air supply volume. **Figure 11.** The relationship between the dimensionless smoke layer height and the dimensionless air supply volume.

**3.7/h\* ×exp(0.027N) Figure 11.** The relationship between the dimensionless smoke layer height and the dimensionless Based on the above results, the empirical formula for the calculation of the dimensionless smoke layer height under the experimental conditions can be given: Based on the above results, the empirical formula for the calculation of the dimensionless smoke layer height under the experimental conditions can be given:

sionless smoke layer height under the experimental conditions can be given:

the ignition source as a point source heat source. Since the object we study is the development of the smoke plume outside the combustion area, we do not consider the combustion

$$H^\* = \frac{6.37 \times \exp(0.028N) + 50.79h^\*}{(1.9h^\* + 0.7) \times N} \tag{7}$$

Based on the above results, the empirical formula for the calculation of the dimen-

<sup>∗</sup> <sup>=</sup> 6.37 × (0.028) + 50.79ℎ<sup>∗</sup>

In the study of enclosed compartment smoke filling, predecessors usually regarded

the ignition source as a point source heat source. Since the object we study is the development of the smoke plume outside the combustion area, we do not consider the combustion

(1.9ℎ∗ + 0.7) × (7)

*3.4. Prediction of Smoke Layer Height in the Engine Room* 

### *3.4. Prediction of Smoke Layer Height in the Engine Room Fire* **2023**, *6*, 16 10 of 14

In the study of enclosed compartment smoke filling, predecessors usually regarded the ignition source as a point source heat source. Since the object we study is the development of the smoke plume outside the combustion area, we do not consider the combustion process during the fire and the changes in the smoke composition caused by chemical reactions. In actual engine room fire, there is no clear boundary between the smoke layer and the air layer. In the theoretical analysis, the transition layer in the smoke filling process is usually ignored, and only the upper smoke layer and the lower air layer are considered. In addition, this paper assumes that the effect of mechanical air supply on smoke deposition is mainly reflected in the following two aspects: the promotion of mechanical air supply to the fire source; the mechanical air supply promotes the mass mixing of the smoke layer and the lower air layer. process during the fire and the changes in the smoke composition caused by chemical reactions. In actual engine room fire, there is no clear boundary between the smoke layer and the air layer. In the theoretical analysis, the transition layer in the smoke filling process is usually ignored, and only the upper smoke layer and the lower air layer are considered. In addition, this paper assumes that the effect of mechanical air supply on smoke deposition is mainly reflected in the following two aspects: the promotion of mechanical air supply to the fire source; the mechanical air supply promotes the mass mixing of the smoke layer and the lower air layer. Based on the above assumptions, we choose a physical model with the same size and

Based on the above assumptions, we choose a physical model with the same size and structure as the experimental cabin. It is assumed that the communication between the interior of the cabin model and the outside world is only through the air supply vents and exhaust vents, and there are no other openings. If the smoke is used as the control body, as shown in Figure 12, the control body always obeys the mass conservation equation during the smoke settling process. There are three main aspects in the process of quality change in the control body: (1) The smoke generated at the fire source enters the control body to increase the quality of the smoke in the control body; (2) the fresh gas provided by the mechanical air supply enters the smoke layer; (3) the upper mechanical exhaust port removes part of the smoke. The main energy transfer process in the cabin can be assumed to be that the heat generated by the fire source in the cabin is divided into three parts: a part of the heat is used to heat the gas in the cabin; a part of the heat is dissipated outward through the wall; a part of the heat is dissipated outward through the smoke outlet. structure as the experimental cabin. It is assumed that the communication between the interior of the cabin model and the outside world is only through the air supply vents and exhaust vents, and there are no other openings. If the smoke is used as the control body, as shown in Figure 12, the control body always obeys the mass conservation equation during the smoke settling process. There are three main aspects in the process of quality change in the control body: (1) The smoke generated at the fire source enters the control body to increase the quality of the smoke in the control body; (2) the fresh gas provided by the mechanical air supply enters the smoke layer; (3) the upper mechanical exhaust port removes part of the smoke. The main energy transfer process in the cabin can be assumed to be that the heat generated by the fire source in the cabin is divided into three parts: a part of the heat is used to heat the gas in the cabin; a part of the heat is dissipated outward through the wall; a part of the heat is dissipated outward through the smoke outlet.

**Figure 12.** Schematic diagram of the smoke stratification in the engine cabin. **Figure 12.** Schematic diagram of the smoke stratification in the engine cabin.

Therefore, its conservation equation can be expressed as: Therefore, its conservation equation can be expressed as:

$$
\dot{m}\_p + \dot{m}\_b - \dot{m}\_c = \frac{d\rho\_s V\_u}{dt} \tag{8}
$$

(9)

where: where:

ሶ : the mass flow rate of the smoke entering the upper control body from the fire source area; ሶ : the mass flow rate of the air entering the smoke layer from other areas; . *mp*: the mass flow rate of the smoke entering the upper control body from the fire source area; . *m<sup>b</sup>* : the mass flow rate of the air entering the smoke layer from other areas;

ሶ : the mass flow rate of the smoke discharged by the mechanical exhaust; . *m<sup>e</sup>* : the mass flow rate of the smoke discharged by the mechanical exhaust;

௨

௦: the smoke density, which is assumed to be constant and has the same value with air; ௨: the volume of the smoke layer. *ρs* : the smoke density, which is assumed to be constant and has the same value with air; *Vu*: the volume of the smoke layer.

The volume change rate ௨ of the control body is the volume of the micro-element above the control body. Since the thickness of the micro-element is infinitely small, the volume of the entire micro-element can be calculated according to the volume of the cube, that is, ௨ = ௭௨, where ௨ is the smoke layer thickness, ௭ is the cross-sectional area of the cabin at the lower boundary of the smoke layer. The volume change rate *dV<sup>u</sup>* of the control body is the volume of the micro-element above the control body. Since the thickness *dz* of the micro-element is infinitely small, the volume of the entire micro-element can be calculated according to the volume of the cube, that is, *dV<sup>u</sup>* = *AzdZu*, where *Z<sup>u</sup>* is the smoke layer thickness, *A<sup>z</sup>* is the cross-sectional area of the cabin at the lower boundary of the smoke layer.

> <sup>=</sup> ሶ + ሶ − ሶ ௭௦

The Formula (9) can be rewritten as:

The Formula (9) can be rewritten as:

$$\frac{dZ\_{\rm u}}{dt} = \frac{\dot{m}\_p + \dot{m}\_b - \dot{m}\_c}{A\_z \rho\_s} \tag{9}$$

During the combustion process, the hot smoke entrains the surrounding air during the upward movement to form a plume. In the plume theory, there is a law of conservation of mass between the amount of smoke and the entrained air, so the amount of smoke generated depends on the amount of entrained air. In this model, Zukoski's axisymmetric plume model is mainly used, as shown in the following formula.

$$
\dot{m}\_p = 0.21 \left(\frac{\rho\_0 g}{\mathcal{C}\_P T\_0}\right)^{1/3} \dot{\mathcal{Q}}\_\mathbb{C}^{1/3} \mathcal{Z}^{5/3} = \mathcal{C} \dot{\mathcal{Q}}\_\mathbb{C}^{1/3} \mathcal{Z}^{5/3} \tag{10}
$$

where:

*ρ*0: air density;

*g*: gravitational acceleration;

*CP*: specific heat capacity at constant pressure;

*T*0: the environment temperature; .

*Qc* : the convective part of ignition power;

*Z*: The height of the plume away from the fire surface.

Based on previous studies, mechanical ventilation has a promoting effect on the power of the fire source. Therefore, combined with Equation (1), we can assume that the mass loss rate . *m* of the fire source can be written as:

$$
\dot{m} = f(V\_{\rm s}) \tag{11}
$$

Therefore, the convection part . *Q<sup>c</sup>* of the fire power can be written as:

.

$$
\dot{Q}\_c \approx 0.7 \dot{Q} = 0.7 \chi \cdot f(V\_s) \cdot \Delta H\_c \tag{12}
$$

where: .

*Q*: total heat release rate of the fire source; *χ*: combustion efficiency;

∆*Hc*: heat of combustion.

In addition, the fresh gas provided by the mechanical ventilation is not only entrained by the fire source, but the gas in other areas will be mixed with the upper layer smoke due to the gas flow. It is assumed that the proportion of the gas mixed with the upper layer smoke in the mechanical supply air volume is *ψ*. For different air supply heights, *ψ* is different; thus, it can be assumed that *ψ* = *g*(*h*), then:

$$
\dot{m}\_{\flat} = \psi \dot{m}\_{\text{s}} = \mathcal{g}(h)\dot{m}\_{\text{s}} = \mathcal{g}(h)\rho\_{\text{s}}V\_{\text{s}} \tag{13}
$$

During the process of smoke filling, the actual cross-sectional area of the cabin is constantly changing. To give a suitable engineering calculation model, it is assumed that the cross-sectional area of the cabin decreases linearly from top to bottom, as shown in the above figure, so the cross-sectional area of the cabin can be written as:

$$A\_z = kz + b \tag{14}$$

The values of *k* and *b* in the formula are determined by the size of the cabin. By substituting Equations (11), (13)–(15) into Equation (10), and introduce the height *Z<sup>t</sup>* of the smoke layer, we have:

$$-\frac{dZ\_t}{dt} = \frac{dZ\_u}{dt} = \frac{\mathcal{C}(0.7\chi \cdot f(V\_s) \cdot \Delta H\_\varepsilon)^{1/3} Z\_t^{5/3} + g(h)\rho\_s V\_s - \dot{m}\_\varepsilon}{(kZ\_t + b)\rho\_s} \tag{15}$$

Therefore, for the prediction of the height of the smoke layer at different air supply air volumes and air supply port heights, there are:

$$\begin{cases} -\frac{d\underline{Z}\_t}{dt} = \frac{d\underline{Z}\_u}{dt} = \frac{\mathbb{C}(0.7\chi f(V\_s) \cdot \Lambda H\_t)^{1/3} \underline{Z}\_t^{5/3} + \mathfrak{g}(h)\rho\_s V\_b - \dot{m}\_c}{(k\underline{Z}\_t + b)\rho\_s} \\ \qquad \qquad \qquad \qquad \qquad \dot{\underline{Z}}\_t = \dot{\underline{H}} \end{cases} \tag{16}$$

Based on the results of the experiment and the above theoretical analysis, the results of the smoke layer height prediction are given below. Based on the results obtained from the dimensionless mass loss rate, namely, Equations (1) and (2), the fire source power can be written as: *Fire* **2023**, *6*, 16 12 of 14 the dimensionless mass loss rate, namely, Equations (1) and (2), the fire source power can

$$\dot{Q}\_c \approx 0.7 \chi \cdot \dot{m} \cdot \Delta H\_c = 0.7 \chi \cdot \left( -204.54 \cdot e^{-V\_s/0.37} + 7.11 \right) \cdot \Delta H\_c \tag{17}$$
 
$$\begin{bmatrix} \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \end{bmatrix}$$

The above formula can be applied to the fire source power under different air volume conditions to determine the height of the smoke layer. Formula (17) can be written as: ≈ 0.7 ∙ ሶ ∙ = 0.7 ∙ (−204.54 ∙ ିೞ/.ଷ + 7.11) ∙ (17) The above formula can be applied to the fire source power under different air volume

conditions to determine the height of the smoke layer. Formula (17) can be written as:

$$\begin{cases} \frac{dZ\_t}{dt} = -\frac{\mathcal{C}\left[0.7\chi\left(-204.54\cdot\text{e}^{-\frac{V\_0}{0.25}} + 7.11\right)\cdot\Delta H\_c\right]^{1/3}z^{\frac{5}{3}}}{\text{when } t=0, Z\_t=1.9 \end{cases} \tag{18}$$

where: where:

be written as:

൞ ௧

*C*: according to Zukoski's model can be calculated, *C* = 0.076432; *k*: *k* depends on the compartment structure, *k* = 6.7; *b*: *b* depends on the compartment structure, *b* = 51; C: according to Zukoski's model can be calculated, C = 0.076432; k: k depends on the compartment structure, k = 6.7; b: b depends on the compartment structure, b = 51;

*ψ*: the mixing rate, a function of the air supply port height h, *g*(*h*) = −0.25*e* <sup>−</sup> *<sup>h</sup>* 0.64 + 0.76. <sup>ψ</sup>: the mixing rate, a function of the air supply port height h, (ℎ) = −0.25ି బ.లర + 0.76.

Put different air volume and air inlet height into the above formula, and iteratively solve based on the second-order Runge–Kutta method. The comparison results between the experimental value and the theoretical value are shown in Figure 13. Compared with the experimental results, it is found that within the range of the experimental air volume, the theoretical prediction results of smoke layer height are consistent with the experimental results, and the relative error is less than 11%. It can be considered that the theoretical prediction model given in this paper has a good prediction effect. Put different air volume and air inlet height into the above formula, and iteratively solve based on the second-order Runge–Kutta method. The comparison results between the experimental value and the theoretical value are shown in Figure 13. Compared with the experimental results, it is found that within the range of the experimental air volume, the theoretical prediction results of smoke layer height are consistent with the experimental results, and the relative error is less than 11%. It can be considered that the theoretical prediction model given in this paper has a good prediction effect.

**4. Conclusions** 

**Figure 13.** Comparison of predicted and experimental values of smoke layer height under different supply air volume conditions. **Figure 13.** Comparison of predicted and experimental values of smoke layer height under differentsupply air volume conditions.

of different air supply volume and air supply inlet heights were studied by experiment method. The theoretical analysis was carried out to predict the height of the cabin fire smoke layer under different air supply conditions and verified the validity of the model.

In this paper, experimental and theoretical research were carried out on the height of

(1) The fire experiment was carried out in the small-scale engine room. Experimental results show that the mass loss rate of the fire source increased approximately exponentially with the increase in the supply air volume. Combined with the experimental data and previous data, the empirical formulas of mass loss rate and air supply volume are

The main conclusions are as follows:
