Modeling a multiple inlet piping system

[MexChemE]

Hello, PF!

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. v₁, P₁, v₄ and P₄ can be measured on the field. P₂ and P₃ 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:
$$\sum \left[ \left( \frac{v_1^2}{2} + gh_1 + \frac{P_1}{\rho_1} \right) w_1 \right] - \sum \left[ \left( \frac{v_2^2}{2} + gh_2 + \frac{P_2}{\rho_2} \right) w_2 \right] = - W_m + E_c + E_v$$
$$\sum \left[ \left( \frac{v_1^2}{2} + gh_1 + \hat{H}_1 \right) w_1 \right] - \sum \left[ \left( \frac{v_2^2}{2} + gh_2 + \hat{H}_2 \right) w_2 \right] = - W_m - Q$$
Where subscript 1 represents inlets and subscript 2 represents outlets. There's no pump inside 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!

 

[Nidum]

I am having trouble understanding how the arrangement that you have drawn can work . You seem to have flows originating and disappearing almost at random and with no obvious system of pressure differences to cause any steady state flow at all .

Please post a revised diagram and some more descriptive information so that we can understand this problem properly .

Actually doing calculations for pipe work and tank systems is not usually very difficult and we can probably help you with these once we have enough information to work with .

 

[MexChemE]

I am having trouble understanding how the arrangement that you have drawn can work . You seem to have flows originating and disappearing almost at random and with no obvious system of pressure differences to cause any steady state flow at all .

Please post a revised diagram and some more descriptive information so that we can understand this problem properly .

Actually doing calculations for pipe work and tank systems is not usually very difficult and we can probably help you with these once we have enough information to work with .

There's a pump upstream from point 1 and another one downstream from point 4. As I mentioned before, both velocity and pressure can be measured at points 1 and 4. What I need to find is the velocity at points 2 and 3. Point 3 seems easier to analyze, as I could model the liquid collector at the bottom of condenser 1 as a tank discharging from the bottom. I could do the same for point 2 (condenser 2), but I don't know how the flow from point 1 will affect it, or if there's any chance it could cause flooding in condenser 2.

 

[Chestermiller]

You would use the first equation if you want to analyze the flow part of the problem. If you wanted to get the temperature changes or the heat flows, you would use the second equation (or, better yet, the difference between the two equations).