To maintain pump power requirements per unit flow rate below an acceptable level, operation of the oil pipeline of Problem 8.63 is subject to the constraint that the oil exit temperature T_m,o exceed 110°C. For the values of T_m,i, T_s, D, t_i, z, L, and ki prescribed in Problem 8.63, operating parameters that are variable and affect T_m,o are the thermal conductivity of the soil and the flow rate of the oil. Depending on soil composition and moisture and the demand for oil, representative variations are 0.25 ≤  k_s ≤ 1.0 W/m · K and 250 ≤ m_o ≤ 500 kg/s. Using the properties prescribed in Problem 8.63, determine the effect of the foregoing variations on T_m,o and the total heat rate q. What is the worst case operating condition? If necessary, what adjustments could be made to ensure that T_m,o ≥ 110°C for the worst case conditions?

Data From Problem 8.63

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 m_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 p_o = 900 kg/m³, c_p,o = 2000 J/kg · K, v_o = 8.5 X 10^-4 m²/s, k_o = 0.140 W/m · K, Pr_o =10⁴; the oil flow rate is m_o = 500 kg/s; and the pipe diameter is 1.2 m.
(a) Expressing your results in terms of D, L, z, t, m_o, T_m,i, and T_s, 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, z = 3 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.