evaporated fuel never contributed to the heat release. In addition, if there exist other miscellaneous gases, neglected here, related to imperfect combustion, the value of imperfectness could increase more.

The tunnel flows are based on the buoyancy due to heat sources. The convective heat released from flames contributes the buoyant flows, as mentioned by Schneider $ ^{23)} $ who studied a large open pool fire. Heselden assumed the 25% radiation loss for the flame with 3–20 MW. May et al. $ ^{24)} $ estimated the radiation heat loss of 16.4%. Here the radiation loss is assumed to be 20% of the evaporated fuel. Thus it is assumed that in total 65% of the fuel was relevant to the convective heat, namely about 700 kW. Previously, it was reported $ ^{25)} $ that the portion of convective heat is generally 40% of the total heat release rate, but it seems too low judging from the report of Koseki et al. $ ^{26)} $ indicated the value of 70% for open pool fires. Above all, the previous numerical studies $ ^{18,27)} $ have used the value of 100% of the total heat release.

The radiation thermally plays a great role in tunnel fires, but it is quite difficult to calculate the radiative transfer among flame, smoke, ceiling, wall and floor. Heselden assumed the temperature of $ 1000^{\circ} $ C in the layer above the flame. However, since the heated floor and walls of the tunnel must affect the flows in tunnels, the floor and walls heated by radiation was taken into account. The complex radiation transfers between flame and ceiling, and between ceiling and floor were not introduced in this calculation. It was left as the problem in next studies since the dense black smoke absorbs the radiation considerably. Here only the radiation transmitted from flame to floor and walls was examined.

It was assumed that the radiant energy due to a flame was focused at a fictitious point $ (P) $ , located on the central axis of the flame and at 0.8 m above the floor. Thus, calculations of the complex configuration factor between flame and solid boundaries become much easier. Totally 100 or 200 kW (20% of total heat release) of radiant heat $ (\Gamma) $ is transmitted from the point source to the solid boundaries such as walls and floor which were divided into grids. The objective to employ this method is to evaluate the effect of the heated solid boundaries due to radiation. Radiation per unit surface area of each grid $ (\gamma) $ was defined by the function of (1) the inverse of the square of the distance $ (r) $ connecting the point P and the center C of the grid surface and (2) the angle $ (\theta) $ between the normal direction of each grid surface and the line connecting points P and C. Thus the relationships become $ \gamma = k \cos \theta / r^{2} $ and $ \Gamma = \sum \gamma_{i} S_{i} $ where k is the constant, $ S_{i} $ is the each grid surface area and i is the grid number.

The heat loss through the ceiling, wall and floor is thermally important in tunnel fires. Kitahara et al. $ ^{[16]} $ used the heat transfer coefficient of $ 5.8 \, W/m^{2}K $ , but Kumar et al. $ ^{[18]} $ employed $ 5-40 \, W/m^{2}K $ . Van de Leur et al. $ ^{[28]} $ , who studied the two-dimensional stratified smoke flow in a corridor, used the value of $ 12 \, W/m^{2}K $ . Here the value by Kitahara et al. was used.

Further Heselden mentioned that about 50% of the combustion can take place in the vertical flame and the remainder must occur in the horizontal hot gas layers, but he carried out the calculations as though the combustion was complete in the vertical flame above the liquid spill. Also in our calculations, the flame region was volumetric and limited vertically above the fuel tray, but the base size of the flame was taken to be twice the tray size since in experiments the flame was burning far outside the fuel tray. The flame height was taken to reach the ceiling. The burning flame extending horizontally was also left as the next problem.