sidering fluid flow at steady state. The fluid entering the main branch is of density po, while the

connecting branch brings in a different fluid of density pt. The uniform flow velocity entering the

main branch is Uo, while the uniform flow velocity entering the connecting branch is U 3Uo. The

two liquids do not react, but mix together to lead to an effective density that can be represented

in terms of proportionality of mass flow rates as follows:

Pe

Polmol + Pelme

mol + m₂

Part a: Sketch a suitable control volume for the problem. Mark out the outward normal directions

along all major control volume faces. Indicate a coordinate system as appropriate.

Part b: Compute the mass flow rate at the entry point along the main tube mo.

Part c: Compute the mass flow rate at the entry point along the connecting tube m.

Part d: Use the above results to compute the effective density at the exit point along the main

tube pe in terms of the scaling ratios a and 3.

Part e: State the Reynolds Transport Theorem for Mass Balance for this model. Mention what

key assumptions you have made to arrive at your final solution.

Part f: Combining your findings from the parts b-e, obtain an expression for the uniform exit flow

velocity Ue./nProblem 4: Mass balance with different fluids

30 points

A T-connection arrangement at a flow handling line in an industrial manufacturing plant is sketched

as below.

Main Branch Tube

Connector Tube

Exit Area

Same As Main

Inlet Area

The main branch tube is cylindrical with a circular cross-sectional radius Ro, while the connecting

branch tube is cylindrical with a circular cross-sectional radius R = Ro. Assume we are con-