The problem of heat losses from a fluid moving through a buried pipeline has received considerable attention. Practical applications include the trans-Alaska pipeline, as well as power plant steam and water distribution lines. Consider a steel pipe of diameter D that is used to transport oil flowing at a rate _o through a cold region. The pipe is covered with a layer of insulation of thickness t and thermal conductivity k_i and is buried in soil to a depth z (distance from the soil surface to the pipe centerline). Each section of pipe is of length L and extends between pumping stations in which the oil is heated to ensure low viscosity and hence low pump power requirements. The temperature of the oil entering the pipe from a pumping station and the temperature of the ground above the pipe are designated as T_m,i and T_s, respectively, and are known. Consider conditions for which the oil (o) properties may be approximated as  
Pr_o = 10⁴; the oil flow rate is _o = 500 kg/s; and the pipe diameter is 1.2 m.

(a) Expressing your results in terms of D, L, z, t, _o, T_m,i, and Ts, as well as the appropriate oil (o), insulation (i), and soil (s) properties, obtain all the expressions needed to estimate the temperature Tm,o of the oil leaving the pipe.

(b) If T_s = 40°C, T_m,i = 120°C, t = 0.15 m, k_i = 0.05 W/m∙ K, k_s 0.5 W/m∙ K, z³ m, and L = 100 km, what is the value of T_m,o? What is the total rate of heat transfer q from a section of the pipeline?

(c) The operations manager wants to know the tradeoff between the burial depth of the pipe and insulation thickness on the heat loss from the pipe. Develop a graphical representation of this design information.