Consider the oil flowing in a pipeline Due to high viscosity

Consider the oil flowing in a pipeline. Due to high viscosity, the effect of friction tends to warm up the fluid as it advances through the pipe. The heat generated by this effect is offset by the cooling provided all along the pipe wall, which is at temperature Ts. The flow is hydrodynamically and thermally fully developed and laminar.

Solution

Integrating energy equation of fluid, we get

k*(dT/dr) + (1/3)(U*((dU/dr)³)) = C

where C is constant.

Again integrating above equation, we get temperature distribution,

k*T(r) + (1/12)*(U²)*(dU/dr)⁴ = C*r +C₁

where C₁ is constant

rearranging above terms,

T(r) = [C*r +C₁ - (1/12)*(U²)*(dU/dr)⁴ ]*(1/k)