*2.1. Governing Equations*

The flow of smoke in a building fire exists in the form of turbulent flow. Several conservation laws should be followed in the process of turbulent flow, including conservation of mass, conservation of momentum, conservation of energy and conservation of components. Therefore, a turbulence model should be established when studying building fire, and so should the computer simulations. Various conservation laws are expressed in the form of governing equations, which are introduced below [33].

(1) Equations for conservation of mass

$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0 \tag{1}$$

where *ρ* is the air density, kg/m<sup>3</sup> ; *t* is time, s; *u*, *v*, *w* are the vectors in direction *x*, *y*, *z*.

(2) Equations for conservation of momentum

$$\frac{\partial(\rho u)}{\partial t} + \frac{\partial(\rho uu)}{\partial \mathbf{x}} + \frac{\partial(\rho uv)}{\partial y} + \frac{\partial(\rho uw)}{\partial z} = \frac{\partial}{\partial \mathbf{x}} \left( \mu \frac{\partial u}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial u}{\partial y} \right) + \frac{\partial}{\partial z} \left( \mu \frac{\partial u}{\partial z} \right) - \frac{\partial p}{\partial \mathbf{x}} + \text{S}\_{\mathbf{u}} \tag{2}$$

$$\frac{\partial(\rho v)}{\partial t} + \frac{\partial(\rho vu)}{\partial x} + \frac{\partial(\rho vv)}{\partial y} + \frac{\partial(\rho vw)}{\partial z} = \frac{\partial}{\partial x} \left( \mu \frac{\partial v}{\partial x} \right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial v}{\partial y} \right) + \frac{\partial}{\partial z} \left( \mu \frac{\partial v}{\partial z} \right) - \frac{\partial p}{\partial y} + S\_v \tag{3}$$

$$\frac{\partial(\rho w)}{\partial t} + \frac{\partial(\rho wu)}{\partial x} + \frac{\partial(\rho wv)}{\partial y} + \frac{\partial(\rho ww)}{\partial z} = \frac{\partial}{\partial x} \left( \mu \frac{\partial w}{\partial x} \right) + \frac{\partial}{\partial y} \left( \mu \frac{\partial w}{\partial y} \right) + \frac{\partial}{\partial z} \left( \mu \frac{\partial w}{\partial z} \right) - \frac{\partial p}{\partial z} + S\_w \tag{4}$$

where *ρ* is the air density, kg/m<sup>3</sup> ; *t* is time, s; *u*, *v*, *w* are the vectors in direction *x*, *y*, *z*; *p* is the pressure of fluid microelement, Pa; *Su*, *Sv*, *S<sup>w</sup>* are generalized source term.

(3) Equations for conservation of energy

$$\frac{\partial(\rho T)}{\partial t} + \frac{\partial(\rho uT)}{\partial x} + \frac{\partial(\rho vT)}{\partial y} + \frac{\partial(\rho wT)}{\partial z} = \frac{\partial}{\partial x} \left( \frac{k}{c\_p} \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( \frac{k}{c\_p} \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( \frac{k}{c\_p} \frac{\partial T}{\partial z} \right) + \mathcal{S}\_T \tag{5}$$

where *T* is temperature, K; *k* is heat transfer coefficient; *c<sup>p</sup>* is specific heat capacity, kJ/(kg·K); *S<sup>T</sup>* is viscous dissipation term.

(4) Equations for conservation of component

$$\frac{\partial(\rho c\_s u)}{\partial \mathbf{x}} + \frac{\partial(\rho c\_s v)}{\partial y} + \frac{\partial(\rho c\_s w)}{\partial z} = \frac{\partial}{\partial \mathbf{x}} \left( D\_s \frac{\partial \rho c\_s}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( D\_s \frac{\partial \rho c\_s}{\partial y} \right) + \frac{\partial}{\partial z} \left( D\_s \frac{\partial \rho c\_s}{\partial z} \right) + S\_s \tag{6}$$

where *c<sup>s</sup>* is the volume concentration of component *S*; *D<sup>S</sup>* is the mass of the component *S* generated by the chemical reaction of unit time and unit volume

(5) Boundary condition

$$-k\_s \frac{\partial T\_s}{\partial t} = \dot{\dot{q}}\_c'' + \dot{\dot{q}}\_r'' \tag{7}$$

where *qc*" is convective heat transfer, W; *qr*" is radiant heat, W.
