# Pipe network analysis: static pressure of a closed-loop pipe

I've learned fluid dynamics but does not have much knowledge about its real-world applications.
My question is about the static pressure of a closed-loop pipe.

[Fig. 1. Pipe network. Source: Wikipedia Pipe_network_analysis]

According to the pipe network analysis, two conditions are satisfied for a steady-state closed-loop pipe flow.

1. At any junction, the total flow into a junction equals the total flow out of that junction.
2. Between any two junctions, the head loss is independent of the path taken. This is equivalent mathematically to the statement that on any closed loop in the network, the head loss around the loop must vanish.

I can understand above fundamental laws for the pipe network analysis.
However, I have difficulty analyzing the static pressure of each junction (not the head loss between two junctions).

Let a pump is located between J3 and J6 of Fig. 1.
Then following conditions will be hold:

1. $$Q_{in}=Q_{out}$$
2. $$P_{6} - P_{3} =$$ (actual head loss of the pump)
3. $$(P_{6} - P_{3})$$ and $$Q_{in}=Q_{out}$$ are related by the pump characteristic curve.

Above five (2+3) conditions just tell about the pressure difference, not the absolute pressure value itself.
If the pump is turned off (i.e. no flow and no head loss) and there is no valve, each joint would have a same absolute pressure value which is determined by the total volume and mass of the fluid filled within the pipe network.

However, what will be the absolute value of $$P_{3}$$ and $$P_{6}$$ when the pump is operated?
Does the absolute pressure of each joint maitain a steady value for a steady pipe network?
If so, which factor determines the steady-absolute pressure value of each joint?
Or, can the steady-absolute pressure value vary depending on transient fluctuations during the pump turn-on period?
Or, can the absolute pressure value vary even for a steady pip network?

• The analysis is analogous to an electrical network -- network theorems like Norton or Thevenin (and other more modern ones) can be applied - writing out system of equations. Sometimes it can be made easier by transforming the network into a series of "voltage dividers", sometimes some algebraic manipulation is unavoidable. If flow is laminar, the analogy to impedance is just like a resistor. Otherwise the impedances are also nonlinear, but that is completely separate from the network analysis, which is done in terms of a-priori-unknown "impedance" of segments: Z_1,2 = (P_2 - P_1) / Q_1,2 etc Aug 7, 2021 at 14:43
• There is nothing in your math model to determine the absolute pressure, and (within practical limits) the absolute pressure makes no difference to the flow. A real world closed-loop system (e.g. a domestic heating system with a pump, boiler, and radiators) has an external connection that is opened to set the absolute pressure and closed when the system is running - and also a device to monitor the absolute pressure and shut the system down if it goes outside its safe operating limits. Aug 7, 2021 at 15:06
• @PeteW Thank you for your comment. Yes, this question is somewhat related to an electrical network. In case of a closed-loop electrical circuit without a ground, it is possible to calculate the voltage difference between two junctions. The term 'absolute voltage' is uncertain because the voltage meter requires two measuring point. Aug 7, 2021 at 15:35
• Then, what if we measure the voltage difference between a junction and ground (like a gauge pressure meter with one-measuring point)? There is no constraint to determine this voltage difference. If so, will this value be continuosly changed? Or will it maintain a fixed value? My question is a fluid-dynamics version of this question. Aug 7, 2021 at 15:35
• @alephzero Thank you for your comment. I agree that the absolute pressure does not affect the fluid dynamics and only related to the safety problem. This question is originated from this safety problem. In my system, the absolute pressure value fluctuates during the pump operation. I want to identify this is a common operation or a failure that can cause an extreme high pressure situation. Aug 7, 2021 at 15:35

Suppose a simple pipe loop with only one path from the pump outlet to the pump inlet.

[Assumption] Neglect effects of other system components and pipe bending.
[Def.] Let the stream direction as $$s$$.
[Def.] Let the total pipe path length as $$L$$.
[Def.] Let the total mass of the fluid filled within the pipe as $$m$$.

There are three variables to be determined: $$P_{in}$$, $$P_{out}$$, and $$Q$$.
($$P_{out}$$ and $$P_{in}$$ are the pressure values of pump outlet and inlet flow, respectively.)
Three equations are needed to solve this problem.

1. [Pump characteristic] $$P_{out} - P_{in} = f_{pump} ( Q )$$.
2. [System curve] $$P_{out} - P_{in} = f_{pump} ( Q ) = k L Q^{n}$$ where $$k$$ is the friction factor.
3. [Mass conservation] $$m = \int_{0}^{L} \rho(P,T)Ads = \int_{0}^{L} \rho(P_{in} + k s Q^{n},T)Ads$$

There are two issues related to the Equation 3.

1. Equation 3 assumes a compressible flow (otherwise, it is meaningless). The compressible effect also has to be applied to other two equations.
2. Bulk modulus of water is quite high. The compressible flow assumption does not make much sense for water.
• (2) is wrong, P_out = P_in - kLQ^n put another way flow is alsways high pressure to low & flow causes pressure loss, not gain. (1) looks ok, (3) I didn't check en detail, water is incompressible for our purposes (even gases can be treated as incompressible when the pressure loss is low) so it should be far simpler.
– mart
Sep 7, 2021 at 13:23
• @mart Thanks, I missed the sign. In case of the compressible flow assumption, I agree that it does not make much sense but there is no other equation I can think. Sep 8, 2021 at 1:13
• mass conversation is accounted for for all nodes with Q_in = Q_3 + Q_9 (one equation for each node plus one for the network)
– mart
Sep 8, 2021 at 12:44
• @mart I agree that the name 'mass conservation' in this answer caused some misunderstanding. Q_in=Q_3+Q_9 stands for the mass conservation for each node; while (3) stands for the mass conservation for the total fluid within the closed-loop pipe. Sep 14, 2021 at 3:43
• Mass conversation of the whole network is counted for by Q:in = Q_out. What's the use of the very complicated integration at [3]?
– mart
Sep 14, 2021 at 6:53