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I’m trying to model a real piping system, which has multiple inlets and one outlet, so I can’t use Bernoulli’s equation. Instead, I’m planning to use the generalized macroscopic energy balances as shown in BSL, which allow for any number of inlets and outlets. However, first I want to make sure if my implementation of the balances is going to make sense.

I’ve attached a diagram of the system I’m trying to model (please, excuse my drawing skills ). I have two condensers, 2 meters above my reference level, everything else is at height zero. The tank’s inlet and outlet are both at the bottom of it. v_{1}, P_{1}, v_{4}and P_{4}can be measured on the field. P_{2}and P_{3}may be calculated, since both condensers have a liquid collector at the bottom, which could essentially be modeled as a tank. Therefore, I could calculate the pressure at the bottom of the collectors.

Now, BSL proposes both a mechanical energy balance and a total energy balance for the analysis of macroscopic systems, shown here:

[tex]\sum \left[ \left( \frac{v_1^2}{2} + gh_1 + \frac{P_1}{\rho_1} \right) w_1 \right] - \sum \left[ \left( \frac{v_2^2}{2} + gh_2 + \frac{P_2}{\rho_2} \right) w_2 \right] = - W_m + E_c + E_v[/tex]

[tex]\sum \left[ \left( \frac{v_1^2}{2} + gh_1 + \hat{H}_1 \right) w_1 \right] - \sum \left[ \left( \frac{v_2^2}{2} + gh_2 + \hat{H}_2 \right) w_2 \right] = - W_m - Q[/tex]

Where subscript 1 represents inlets and subscript 2 represents outlets. There's no pumpinside the control system, so W_{m}= 0. Since the flow is incompressible, E_{c}= 0, and E_{v}represents head losses due to friction and valves, fittings, etc. As you can see in the diagram, there are significant temperature changes across the system, so I suppose both energy balances should be taken into account.

To summarize in a couple of questions:

1. In order to model my system, given the necessary additional equations such as the continuity equation, is it as simple as just plugging in for inlets and outlets, and solve for the unknown variables (in the same way we model a piping system from point A to B with Bernoulli's equation)?

2. How can I reconcile both the mechanical and total energy balances? Is it valid to use them at the same time to model the same system?

Thanks in advance for any input!

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# Modeling a multiple inlet piping system

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