Questions regarding the calculation of pipeline pressure loss
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A fluid flows out from a high-level tank, where the liquid level remains constant, and then passes through a straight section of pipe equipped with a valve. The pipe diameter d, the equivalent length of the pipe l, the local resistance coefficient of the valve ζ, the flow velocity u, and the frictional resistance coefficient within the pipe λ are known. When calculating pipeline resistance, is the expression for pipeline resistance hf= λl/d*u^2/2+ζ*u^2/2 or hf= λ*l/d*u^2/2+ζ*u^2/2+u^2/2? Personally, I lean towards the first option, because the fluid still possesses a certain amount of kinetic energy after exiting the pipeline, and this kinetic energy should not be included in the pipeline resistance. However, an analysis of a problem in the chemical engineering course materials from East China University of Science and Technology left me confused about pipe resistance; I have attached that part in the image, and I hope the teachers can provide some guidance.
The premise is incorrect. The condition that the resistances in parallel pipelines are equal applies only to parallel pipelines; your problem does not state that it involves parallel pipelines. From the diagram, it appears to be a single branch pipeline, so calculations should be done using the method applicable to branch pipelines, that is, the total mechanical energy should be equal to the sum of the energy losses
Sharing my personal opinion, please correct me if I’m wrong. I agree with your approach of analyzing flow velocity distribution from the perspective of resistance. The formula used in the PPT for analysis from an energy perspective is worth examining; the key question is whether the energies at the outlets of pipelines B and C are equal The energy at any point includes kinetic energy, potential energy, and volume work. The premise for energy conservation is mass conservation; applying energy conservation to branched pipelines is inappropriate~~
The formula for energy conservation is derived from the Bernoulli equation, which was developed using Euler’s method by analyzing a unit mass of fluid. The units of static head, potential head, and dynamic head are all J/Kg or J/N. Therefore, the distribution of flow in branch pipes also applies when formulating the energy conservation equations starting from point O. The height of the liquid level in the container, as well as the diameter and length of the pipes leading up to point O, only affect the energy at point O; they have no impact on the subsequent energy conservation equations. My question is that this textbook contains two problems of the same type, but the solutions provided are not consistent; more importantly, the answers obtained from these two solutions differ. I just want to know which method is unreasonable or what the differences are in the specific applicable conditions of these methods (which I didn’t notice). Below, I’ll post the two questions again.
I think the examples you gave are all correct; both the principles of parallel circuits and branched circuits derive from Bernoulli’s equation. The difference is that in parallel circuits, the pipes converge together, and the flow rate remains constant. At the branching points and at the point of convergence, the flow velocities are necessarily equal, and the changes in pressure across each branch are also the same. Therefore, an equation stating that the resistances are equal can be derived. In fact, the principle of branched circuits is nothing but Bernoulli’s equation. To put it simply, parallel circuits represent a special case of branched circuits. Looking at the two examples you gave, I already responded to the first one: the premise used was incorrect; it should be considered in terms of branch pipelines. For the second question, the premise states that all backpaths are open, which means they converge together. The flow rate is the same as that at point A, and therefore the difference in flow velocity at the convergence point compared to point A is constant, as is the height difference and the pressure difference. Hence, the resistance will also be constant.
In other words, only when the outlets of the two pipes are at the same level can the outlet of those branches be considered as the junction point of a parallel pipeline, and then the assumption that the resistances of all branches in such a parallel pipeline are equal applies. However, if the outlets of the various branches are not at the same level, they do not actually constitute a junction point (due to the difference in elevation). Is that how I should understand it?
There is another question: based on your previous explanations, I understood how to solve the problem that involves applying the law of conservation of energy. But why can’t the problem in Example 1-13, which deals with the analysis of parallel pipelines, be solved using the law of conservation of energy?
In parallel pipelines, what is calculated is the resistance of each branch from its junction point to the convergence point, whereas in branched pipelines, what is calculated are the energy values at each pipe outlet; in other words, their resistance losses are not the same. In 1-13, the pressure loss from B to C is different from the pressure loss when fluid flows from B through C out of the pipeline, and it can also be understood that Pc and Pd within the pipeline are not equal
Could you tell me the book title? I’m learning *ha.
I learned it; that’s great, it helped reinforce the basic knowledge: lol