In areas where the air temperature remains below o°c for prolonged periods of time, the freezing of water in underground pipes is a major concern. fortunately, the soil remains relatively warm during those periods, and it takes weeks for the subfreezing temperatures to reach the water mains in the ground. thus, the soil effectively serves as an insulation to protect the water from subfreezing temperatures in winter. the ground at a cold location is covered with snow pack at -10°c for a continuous period of two months, and the average soil properties at that location are k 0.4 w/m-k and a 0.15 x10m/s. each pipe of the system is 30 m long with a diameter of 10 cm. assuming an initial uniform temperature of 10°c for the ground and pipe surface, determine: a) the minimum burial depth to prevent the water pipes from freezing, and b) the conduction shape factor and steady rate of heat transfer at the minimum burial depth.