Exercise 1: Heat distribution

Part of monitoring a structure is determining its heat distribution; this may help to avoid thermal stress, where a material such as concrete or steel may crack or develop a weakness on account of heat or cold.

This problem concerns a concrete bridge support, shown in figure 1, which has a square cross section. It bottom is anchored in the river bed; its sides are in the water, and its top is exposed to the air.

1. Find appropriate values for the temperature of the river bed, and of the water and air. For this problem, you can assume that the water has the same temperature at every depth.

2. To determine the heat distribution of the bridge support, we shall consider four internal equally spaced "nodes", which are shown on the left in figure 2.

For computing the temperature at each node we can use the "mean value property" which is illustrated on the right; this says that the temperature at a point is the average of the temperature of four equally spaced points: b = (a₁ + a₂ + a₃ + a₄)/4.

Use your values from part 1 to develop a set of equations for the temperature at each of the nodes, and solve those equations.