# Theoretical and Numerical Simulation Research on Fire of Large-Span Spatial Structures
## 3 Temperature Field Analysis of Structure Under Different Fire Conditions
### Axis Temperature and Velocity of Axisymmetric Flame Plume of the Structure
Through the experimental data, [PERSON] obtained the calculation equation of the plume centerline temperature above the position of the virtual point source and the average flame height, as shown in Equations (3) and (4). According to the average flame height, [PERSON] divided the fire plume into two calculation zones. \[z_{0}=0.083\dot{Q}_{\mathrm{c}}^{2/5}-1.02D \tag{3}\]  
\[\Delta T=9.1(\frac{T_{0}}{g\mathrm{c}_{\mathrm{p}}P_{0}^{2}})^{1/3}\dot{Q}_{ \mathrm{c}}^{2/3}(z-z_{0})^{-5/3} \tag{4}\]  
In the formula, \(z_{0}\) is the height of the virtual fire source, m; \(D\) is the diameter or equivalent diameter of the fire source, m.  
(3) [PERSON] model [7, 8]:  
Similar to [PERSON]'s division of the plume into two computational regions, [PERSON] further divided the fire plume into three computational zones according to temperature: continuous flame zone, intermittent flame zone, and smoke plume area. And [PERSON] obtained the calculation equation of plume temperature in the corresponding plume region by means of experimental data and dimensionless method, as shown in Equations (5)-(7):  
Continuous flame zone  
\[\frac{z}{\dot{Q}_{\mathrm{c}}^{2/5}}\leq 0.08,\ \Delta T=28.54\frac{T_{0}}{g} \tag{5}\]  
Intermittent flame zone  
\[0.08<\frac{z}{\dot{Q}_{\mathrm{c}}^{2/5}}\leq 0.2,\ \Delta T=2.23\frac{T_{0}}{g}( \frac{z}{\dot{Q}_{\mathrm{c}}^{2/5}})^{-1} \tag{6}\]  
Smoke plume area  
\[\frac{z}{\dot{Q}_{\mathrm{c}}^{2/5}}>0.2,\ \Delta T=0.747\frac{T_{0}}{g}( \frac{z}{\dot{Q}_{\mathrm{c}}^{2/5}})^{-5/3} \tag{7}\]  
In the McCaffrey plume equations, the properties of the plume parameters were assumed to depend only on the energy release rate, that is, independent of the type of fuel block and combustion properties. In the actual fire, the fuel properties are different, and the radiation loss changes are also different. Therefore, the temperature in the flame of different fuel blocks is also different, but this influence factor has little effect on the continuous flame area. (4) [PERSON]'s formula [38]:  
[PERSON] simulated 120 fire scenarios of cuboid space buildings without considering ventilation, and selected the space point temperature value that conforms to the simplified temperature model. Finally, the practical empirical formula of air temperature rise under large space building fire is regressed, such as Equation (8), which is suitable for the discrete form of elements in the finite element analysis of building structure and is easy to be used in structural fire resistance analysis. \[T(x,z,t)-T_{0}=[(20Q+80)-(0.4Q+3)H+(52Q+598)\times 100/A]\times(1-0.8e^{(- \beta t)}-0.2e^{(-0.1\beta t)})\times\kappa \tag{8}\]In the formula, \(T(x,z,t)\) is the temperature value of a certain position at any time in the large space, \({}^{\circ}\)C; \(T_{0}\) is the ambient temperature, \({}^{\circ}\)C; \(Q\) is the fire source power, mw; \(H\) is the height of large space building, m; \(A\) is the construction area, m\({}^{2}\); for ultra-fast fire, fast fire, medium fire, and slow fire, the \(\beta\) values are 0.004, 0.003, 0.002, and 0.001, respectively. \(\kappa\) is the temperature attenuation coefficient. (5) Strong plume velocity formula [50]:  
Continuous flame zone  
\[\frac{z}{\dot{Q}_{\rm c}^{2/5}}\leq 0.08,\ \frac{u_{0}}{\dot{Q}_{\rm c}^{1/5}}=6.8 (\frac{z}{\dot{Q}_{\rm c}^{2/5}})^{1/2} \tag{9}\]  
Intermittent flame zone  
\[\Delta T=5.0(\frac{T_{0}}{gc_{P}^{2}\rho_{0}^{2}})^{1/3}\dot{Q}_{c}^{2/3}z^{- 5/3} \tag{2}\]0  
Smoke plume area  
\[\Delta T=5.0(\frac{T_{0}}{gc_{P}^{2}\rho_{0}^{2}})^{1/3}\dot{Q}_{c}^{2/3}z^{- 5/3} \tag{2}\]1  
In the formula, \(u_{0}\) is the average temperature on the centerline of the fire plume, m/s.