# Theoretical and Numerical Simulation Research on Fire of Large-Span Spatial Structures
## 4 Modified Two-Zone Temperature Field Model
### Correction of the Model  
In a large space building fire with a top opening, the complex interaction between the fire source and the hot smoke layer makes the smoke temperature change with time complicated. In order to simplify the model, it is assumed that the temperature distribution of the space fire is uniform in the horizontal direction, and the spatial inhomogeneity of temperature is mainly reflected in the vertical direction. According to the temperature distribution characteristics of the above fire, the temperature distribution hypothesis of the smoke layer is given. As shown in Figure 29, ignoring the thermal radiation of the fire source, the temperature of the lower air basically remains unchanged at the initial value, and the increase of the gas temperature at a certain height can only be caused by the arrival of the flue gas, while the temperature in the flue gas increases linearly with height [51; 52; 53; 54]. Assuming that the thickness of the flue gas layer at a certain time is h, in the traditional two-zone model, the temperature of the flue gas layer and the air layer is uniform and the temperature value is discontinuous. In order to simplify the calculation, in the modified two-zone model, the temperature of the air layer is uniform and remains at the initial value. The gas temperature in the flue gas layer increases linearly with height, and the front of the flue gas layer--that is, the boundary point between the flue gas layer and the air layer--is 1 point. The gas temperature is equal to the lower air temperature. According to the assumption of temperature distribution in the figure, after the temperature in the space reaches the maximum, the temperature at any height can be expressed as follows:  
\[\left\{ \begin{array}{l}z\leq H-h\;,\,T(z)=T_{0}\\ z>H-h\;,\,T(z)=az+b\end{array} \right. \tag{35}\]  
According to the assumption of linear distribution, the determination of coefficients a and b only needs the temperature values of two points on the distribution line. The most common--the average temperature of the control body--can be chosen, that is, 2 points; the boundary point between the flue gas layer and the air layer is 1 point. The highest point temperature of the control volume, namely 3 points, can be determined by the values of 1 point and 2 points. According to the physical hypothesis of the region model, because the temperature distribution in the flue gas layer satisfies a linear relationship, the temperature in each  
Figure 29: The temperature distribution model. region is uniform, while the temperature change between adjacent regions is continuous, and the temperature at the height of \(h/2\) is the average temperature of the flue gas layer \(T_{s}\). \[T_{s}=\frac{T_{max}-T_{0}}{2} \tag{36}\]  
For a building fire with a top opening, \(\dot{Q}_{\rm c}-\dot{Q}_{\rm e}\) is the convective heat carried by the plume, and \(\dot{Q}_{\rm e}\) is the heat released from the opening, kW, due to the more heat released from the top opening. According to the law of conservation of energy, the following equation can be obtained:  
\[\dot{Q}_{\rm c}-\dot{Q}_{\rm e}=\dot{m}_{p}c_{p}\Delta T \tag{37}\]  
In the process of natural smoke extraction in indoor fires, if the smoke flow rate and the smoke generation rate (plume entrainment rate) are maintained at the same level, the indoor smoke layer will reach a 'quasi-steady state'. In the quasi-steady state, the interface of the indoor smoke layer will be maintained at a stable height. There is the following:  
\[\dot{m}_{p}=\dot{m}_{e} \tag{38}\]  
\[\dot{Q}_{\rm c}-\dot{Q}_{\rm e}=\dot{m}_{p}c_{p}\Delta T=\dot{m}_{e}c_{p}\Delta T \tag{39}\]  
In the equation, \(\dot{m}_{p}\) is the mass flow rate of plume entrainment, kg/s; and \(\dot{m}_{e}\) is the mass flow rate at the opening, kg/s. When it reaches the quasi-steady state, \(\dot{Q}_{\rm e}\) is a quantity that does not change with time, and the ratio of it to \(\dot{Q}_{\rm c}\) is defined as the heat outflow coefficient \(\chi\).