A long concrete beam is to undergo a thermal test to determine its loss of strength in the event of a building fire. The beam cross section is triangular as shown in the sketch. Initially, the beam is at a uniform temperature of 20°C. At the start of the test, one of the short faces and the long face are exposed to hot gases at 400°C through a heat transfer coefficient of 10 W/(m2 K) and the remaining short face is adiabatic. Produce a graph showing the highest and lowest temperatures in the beam as a function of time for the first 1 hour of exposure. For the concrete properties, use k = 0.5 W/(mK) and n = 5 x 10–7 m2/s. Use a node spacing of 4 cm. and use an explicit difference scheme.

Question

GIVEN

Concrete beam suddenly exposed to hot gases

FIND

(a) Highest and lowest temperatures in the beam as a function of time

SKETCH

Answer 1

The arrangement of nodes and control volumes is shown in the figure to right. Examination of this

figure reveals that we have 7 unique control volumes. We need to develop an energy balance for each

type. To simplify the notation, use the following



For al interior control volumes: i = 2, j = 2, 3, 4; i = 3, j = 2, 3; and i = 4, j = 2, we have for the energy

balance



For the bottom row of control volumes (not corners), j = 1, i = 2, 3, 4, 5, we have



For the bottom row of control volumes (not corners), j = 1, i = 2, 3, 4, 5, we have



For control volumes on the diagonal (i, j) = (2, 5), (3, 4), (4, 3), (5, 2) we have



Finally, for the corners



The system of equations can be solved by the marching procedure. We must keep in mind the

limitation in ?t which gives ?tmax = 800 seconds

The equations were solved using ?t = 10 seconds. A check was performed by hand on each of the

seven unique control volume energy balances. The maximum temperature occurs at i = 6, j = 1, and

the minimum temperature occurs at i = 1, j = 3. The resulting temperature as a function of time is

given below.