A concentric tube heat exchanger for cooling lubricating oil is comprised of a thin-walled inner tube of 25 mm diameter carrying water and an outer tube of 45 mm diameter carrying the oil. The exchanger operates in counterflow with an overall heat transfer coefficient of 60W/m2K and the tabulated average properties given below. If the outlet temperature of the oil is 60oC, plot the temperature as a function of x and determine the total heat transfer, the outlet temperature of the water, and the required length.

The information provided is not sufficient to calculate the exact values of total heat transfer, outlet temperature of the water, and the required length, as some key data such as the inlet temperatures, flow rates of the oil and water, and the specific heat capacities of the fluids are missing. However, I can guide you through the process of how to solve this problem, assuming you have the necessary data.

1. Determine the mass flow rates (m_dot) of water and oil if those are given, or assume sensible values based on typical operational conditions. 2. Use the specific heat capacities (C_p) for water and oil provided in your tabulated data to calculate the heat capacity rates for both fluids: C_water = m_dot_water * C_p_water C_oil = m_dot_oil * C_p_oil

3. Identify the inlet temperatures for both the water and oil. In this case, we know the outlet temperature of the oil (60°C), but the inlet temperatures are required to progress.

4. Calculate the log mean temperature difference (LMTD) for counterflow heat exchanger using the inlet and outlet temperatures: LMTD = (ΔT_1 - ΔT_2) / ln(ΔT_1/ΔT_2) where ΔT_1 and ΔT_2 are the temperature differences at each end of the heat exchanger.

5. The total heat transfer Q can be calculated using the overall heat transfer coefficient (U), the LMTD, and the heat transfer area (A): Q = U * A * LMTD

6. For a thin-walled concentric tube, the heat transfer area can be approximated using the diameter of one of the tubes (typically the inner tube for the water in this case) and the length (L): A ≈ π * D_inner * L

7. To solve for the unknowns, you may have to apply energy balance equations for each fluid: Q = m_dot_water * C_p_water * (T_water_out - T_water_in) = m_dot_oil * C_p_oil * (T_oil_in - T_oil_out)

8. To find the water outlet temperature (T_water_out), you would rearrange the above equation: T_water_out = (Q / (m_dot_water * C_p_water)) + T_water_in

9. The length of the heat exchanger (L) is determined by rearranging the formula for Q above: L = Q / (U * π * D_inner * LMTD)

With the right data, by following the steps above, you can calculate the required values. To plot the temperature as a function of x, you may assume a linear temperature gradient or a more complex profile based on the specific details of the flow and any provided temperature data along the length of the exchanger.