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Two-Dimensional Analysis on Ceiling Jet Temperature Characteristics in a Semicircular Tunnel


In order to reveal the evolution of ceiling jet temperature characteristics beneath curved ceiling in two-dimensional view, experimental and theoretical studies were conducted to analyze the vertical temperature profile and longitudinal temperature attenuation beneath a curved ceiling of a semicircular tunnel. The temperature profile was first measured from experiments carried out in the model-scale semicircular tunnel. The relationship between temperature and the horizontal positions is described by an empirical model. A formula describing the thickness of the thermal boundary layer is proposed, based on the vertical distribution of smoke temperature. An engineering model for the longitudinal attenuation of the gas temperature with different fire plume types at impingement region is established in terms of conservation equations. After validation, model predictions show a good agreement with trend proposed by former literature and the relative error of prediction is found to be within 10%. Two-dimensional prediction for temperature profile beneath semicircle ceiling centerline was proposed. It was noted that two-dimensional prediction model could predict majority of measured data within 20% relative error even though some fluctuation, which could be applied to the fire protection engineering in semicircular tunnel.

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source with different fuels. a N-heptane. b Ethanol. c Methanol

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Figure 13
Figure 14


B :

Wet perimeter of thermal layer (m)

c f :

Fanning friction factor

C 1 , C 2 , C 3 , C 4 , C 5 :

Constant obtained from experiments

C p :

Heat capacity of air (kJ/(kg.K))

h c :

Convective heat transfer coefficient (kW/(m2.K))

h rad :

Radiative heat transfer coefficient (kW/(m2.K))

H :

Ceiling clearance above fire source (m) (In this work, H = R)

l :

Half width of tunnel (m)

L :

Width of thermal boundary layer (m)

L T :

The vertical position from ceiling where ΔT/ΔTmax(x) = 1/e (m)

\(\dot{m}\) :

Mass flow of entrained flow and thermal layer (kg/s)

\(\dot{m}_{e}\) :

Mass flow of entrained flow (kg/s)


Prandtl Number

Q conv :

Convective heat loss from thermal layer to the ceiling (kW/m)

Q rad :

Radiative heat loss from thermal layer to the ceiling (kW/m)

R :

The radius of tunnel (m)

Re :

Reynolds number

Ri :

Richardson number

s :

Cross-section area of thermal boundary layer (m2)

St :

Stanton number (St = hc/(Cpρu))

T e :

Characteristic temperature of entrained layer (K) (In this work, Tt = Te)

T t :

Characteristic temperature of thermal layer (K)

T t,o :

Characteristic temperature of thermal layer at the reference point xo (K)

T w :

Temperature of concrete wall (K)

T :

Ambient temperature (K)

ΔT (x,z) :

Temperature rise at point with x from fire source and z from ceiling (K)

ΔT max (x) :

Maximum temperature rise with the same horizontal distance from fire source (K)

ΔT t :

Characteristic temperature rise of thermal boundary layer (K)(In this work, ΔTtTmax)

ΔT t ,o :

Characteristic temperature rise of thermal boundary layer at reference point (K)

u :

Horizontal flow velocity of thermal layer (m/s)

x :

Horizontal distance from fire source (m)

x f :

Horizontal location of the boundary between region II and III (m)

x o :

Horizontal distance from fire source to the reference point (In this work, xo = 3 m)

x v :

Distance between the virtual origin and the fire source (m)

z :

Vertical position from ceiling (m)

Z max :

The vertical position from ceiling where ΔT/ΔTmax(x) = 1 (m) Zmax=δ(x) in this work

γ :

Coefficients in the Eq. 13

δ (x) :

Thickness of thermal boundary layer at x (m)

ε :

Coefficients in the Eq. 13

ε w :

Emissivity of wall surface

ρ :

Smoke density (kg/m3)


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The work was supported by the National Key Research and Development Program of China (No. 2016YFC0802900) and Graduate Research and Innovation Projects of Jiangsu Province (KYCX21_2446). Also, authors would like to thank the China Scholarship Council (CSC) for the financial support of the first author (Grant No. 202006420046).

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Correspondence to Guoqing Zhu.

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Based on the flow mechanisms of the thermal boundary layer described in the Fig. 5, referring to the former literature [13, 35, 41, 43], governing equations of a differential volume in one-dimensional thermal boundary layer were established. There are several assumptions:

  1. 1.

    Steady HRR;

  2. 2.

    The properties of thermal layer in one-dimensional ceiling jet is stable, such as density, velocity;

  3. 3.

    There would be hardly any mass exchange in smoke layer except for entrainment.

With assumptions mentioned above, governing equations for selected control volume are shown as follows:

Continuity equation,

$$\dot{m}_{e} = \frac{{d\dot{m}}}{dx}$$

Energy conservation equation,

$$\frac{{d(C_{p} \dot{m}T_{t} )}}{dx} = C_{p} \dot{m}_{e} T_{e} - Q_{conv} - Q_{rad}$$

The relationship between dTt and dx is expected to obtained from conservation equations mentioned above. As the boundary for Eq.A1, mass flow equation is expressed,

$$\dot{m} = \rho us$$

As shown in the Eqs. A1A3, \(\dot{m}\) and \(\dot{m}_{e}\) represented the mass flow of thermal layer and entrained flow, respectively. The mass flow of thermal layer is described by the smoke density ρ, horizontal flow velocity of thermal layer u and cross-section area of thermal boundary layer s. In addition, Qconv and Qrad are the convective and radiative heat loss from thermal layer to the ceiling, respectively. Considering that fire duration is short in comparison to thermal inertia of concrete in this work, the temperature of concrete wall Tw far from fire source was assumed as that of ambient air T. As a result, Qconv and Qrad could be written as:

$$Q_{conv} = h_{c} B(T_{t} - T_{\infty } )$$
$$Q_{rad} = \varepsilon \sigma B\left( {T_{t}^{4} - T_{\infty }^{4} } \right) = \varepsilon \sigma B(T_{t}^{2} + T_{\infty }^{2} )(T_{t} + T_{\infty } )(T_{t} - T_{\infty } )$$

Based on the reference [45, 57, 58], radiative heat transfer coefficient would be employed in the Eq. A5, which has the same dimension with convective heat transfer coefficient. The Eq. A5 could be transformed as:

$$Q_{rad} = h_{rad} B(T_{t} - T_{\infty } )$$

Incorporating Eqs. A1A6, a new equation was obtained,

$$C_{p} \frac{{dT_{t} }}{dx}\dot{m} + C_{p} \frac{{d\dot{m}}}{dx}T_{t} = C_{p} \frac{{d\dot{m}}}{dx}T_{e} - h_{c} B(T_{t} - T_{\infty } ) - h_{rad} B(T_{t} - T_{\infty } )$$

To simplify the one-dimensional model, on the basic of assumptions, the characteristic temperature of thermal layer Tt was equal to that of entrained layer Te. Thus Eq. A7 was expressed as follows:

$$- \frac{{dT_{t} }}{{T_{t} - T_{\infty } }} = \frac{{(h_{c} + h_{rad} )B}}{{C_{p} \rho us}}dx$$

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Pan, R., Hostikka, S., Zhu, G. et al. Two-Dimensional Analysis on Ceiling Jet Temperature Characteristics in a Semicircular Tunnel. Fire Technol (2021).

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  • Curved ceiling
  • Temperature profile
  • Thermal boundary layer
  • Longitudinal temperature attenuation
  • Two-dimensional temperature profile