2. Consider the cooling of a pure fluid flowing with a velocity of U. in a circular pipe of radius R and length L. The fluid enters the pipe at temperature To. The flow is turbulent, i.e. we assume that plug flow conditions exist. So, the fluid elements are well-mixed and hence there are no radial velocity and temperature gradients. Temperature of the fluid in the reactor, however, is changing with the axial position (z). The pipe has a cooling jacket through which water is circulated. Heat can be transferred from the fluid at temperature T to the walls of the reactor and from walls to the cooling water at constant temperature Tw. The heat transfer between the fluid and the cooling water is governed by the overall heat transfer coefficient, U, and the heat transfer area, A. Assume that conduction heat transfer along the axial direction is small relative to convection. State any other assumption you make. a) Derive the differential equation to determine the variation of fluid temperature along the axial direction at steady-state. b) Solve the differential equation to determine the temperature of the fluid as a function of axial position. c) For fully developed laminar flow conditions, plug flow assumption is no longer valid and the 72 velocity changes with radial position as u=2 u. 1- where u, denotes the average velocity. R? In this case, the fluid is not well-mixed in the radial direction and radial heat conduction cannot be neglected. Moreover, because the velocity is smaller now due to laminar flow (Re