Worked example: a hot and cold mixing tee

When two streams of the same fluid meet and mix, the blended temperature is not the simple average of the two inlet temperatures. It is the flow-weighted average, so the stream carrying the larger flow pulls the result toward its own temperature. Get that wrong and a tempering or blending design lands at the wrong temperature. This worked example shows the mixing rule in action and adds a second effect on top: the blended stream cools further as it travels to a cold ambient, so the temperature you mix is not the temperature you deliver.

The setup

A hot stream and a cold stream meet at a tee and the combined flow runs to a sink, with heat transfer switched on. All pipe is DN100 schedule 40 steel (102.26 mm internal bore, 0.15 mm roughness):

• Hot inlet: 2 L/s entering at 70 degrees Celsius, through 50 m of pipe to the tee.
• Cold inlet: 3 L/s entering at 15 degrees Celsius, through 50 m of pipe to the tee.
• The tee: a mixing node where the two streams combine.
• Mixed run: 300 m of pipe from the tee to the outlet, losing heat to a 5 degree Celsius ambient through a direct overall coefficient of 5 W/m squared.K.

The working fluid is water. The cold stream carries the larger flow, 3 L/s against 2 L/s, so the mix sits below the midpoint of 42.5 degrees: the two streams blend to about 37 degrees Celsius, and the long run to a cold ambient brings the delivery temperature down to just under 35 degrees.

The physics and the method

This is the Advanced heat-transfer mode, and it shows two distinct mechanisms. At the tee the solver applies an enthalpy balance: the energy carried in by both streams equals the energy carried out, so the mixed temperature is the flow-weighted average of the inlets. Because the cold stream carries more flow, the result lands nearer 15 than 70, at roughly 37 degrees. Along the 300 m run the solver applies a direct overall heat-transfer coefficient to the line, so the temperature decays exponentially toward the 5 degree ambient. The flow and temperature fields are solved together, with an energy balance reported as a residual on every solve.

The direct-U option used here is the simple case where you already know the overall coefficient and apply it directly, rather than building it up from films and wall layers. That keeps the focus on the two ideas worth taking away: enthalpy mixing at the junction, and exponential cooling along the pipe.

What you learn

Solving the example gives you the blended temperature at the tee, the delivery temperature at the outlet, and the heat lost along the mixed run, alongside the flows and pressures. Colour the network by temperature and you can watch the two streams converge at the tee and then cool along the run. The instructive experiment is to change the flow split: send more hot and less cold and the mix climbs toward the hot inlet, confirming that it is flow, not a simple average, that sets the result. This is the calculation behind tempering and blending design, covered in context on the pipe heat loss application page.

Heat transfer is part of the Advanced plan, A$50/month or A$500/year. The glossary explains enthalpy mixing, the overall heat-transfer coefficient and temperature decay along a pipe.

Open this example in FNS and change the hot and cold flow split to see the blended temperature move.