Skip to main content. Fluid Dynamics and Its Applications. Search for:. Flow in Tubes. Learning Objectives Contrast turbulent and laminar flow in constant velocity. Key Takeaways Key Points Viscosity is the resistance of a fluid to flow. Virtually all fluids have viscosity which generally changes as a function of temperature; although different types of fluids exhibit different types of fluid—shear velocity dependencies.
Laminar flow of a fluid is characterized by its flow in parallel layers in which there is no disruption or interaction between the different layers, and in which each layer flows at a different velocity along the same direction. It relates the difference in pressure at different spatial points to volumetric flow rate for fluids in motion in certain cases, such as in the flow of fluid through a rigid pipe. Key Terms viscosity : The property of a fluid that resists the force which tends to cause it to flow.
The value of the number indicates the type of fluid flow. Blood Flow Blood flow is the continuous running of blood through the cardiovascular system, which consists of the vessels and the heart.
Learning Objectives Outline how normal plasma behaves in a mammalian cardiovascular system. The mechanics of the circulation depends on osmotic pressure of plasma. Key Terms systole : The rhythmic contraction of the heart, by which blood is driven through the arteries. As the speed becomes faster eddies start to form and cross the fluid layers. A transition from laminar to turbulent flow develops. At still higher velocities the flow in the core of the pipe becomes turbulent with swirling eddies throughout.
Figure 2 shows where the various flow regions occur at a tank nozzle. The laminar sub layer is always present against the pipe wall. But as the velocity rises the energetic swirling eddies begin to impact more deeply and the sub layer begins to thin. At still higher velocities the sub layer thins further and the taller roughness peaks stick into the turbulent region. This minimises the losses along the pipe. There is a very much greater loss of pressure in turbulent flow.
The pipe system designer has to strike a practical balance between increasing the pipe diameter to reduce energy loss and keeping the diameter small to lower installation costs. Elbows, bends, reducers, branch tees and flanges all cause individual minor pressure losses. When a fluid is forced to change direction, or go around a disruption, eddies are produced. These new twisting eddies interfere with the flow pattern and produce additional pressure losses.
The greatest pressure losses occur at sudden diameter and direction changes. Most of the loss occurs in the downstream eddy wake. When designing a pipe run gradually blend-in changes to the flow pattern. Unlike a liquid a gas is compressible and can be squashed.
When a gas is compressed the density increases — as the pressure is released the density decreases. Gas flowing into a pipe starts at a pressure, temperature and associated density. A lever system ensures the transmission of information and of the negative reaction into the system.
The other two subsystems, the servomotor and the control valve body, are interdependent, their connection being of both physical and informational nature. In the drawing of the figure 13 there is presented a servomotor with a diaphragm, a servomotor that ensures a normally closed state of the control valve. The contact element between the two subsystems is represented by the rod 3. This transmits the servomotor movement, expressed by the rod h displacement, towards the control valve body.
The command pressure of the servomotor, p C , represents the output variable of the electro — pneumatic convertor. The control valve body represents the most complex subsystem within the control valve. This will modify the servomotor race h and, accordingly, the valve plug position in ratio with the control chair. The change of the section and the change of the flowing conditions in the control valve body will lead to the corresponding change of the flow rate. The electro — pneumatic convertor: 1 — electromagnetic circuit; 2 — the pressure — displacement sensor; 3 — permanent magnets; 4 — power amplifier; 5 — the reaction bellows; 6 — articulate fitting; 7 - lever.
The servomotor — control organ subsystem: SM — servomotor; CVB- control valve body; 1 - resort; 2 — rigid diaphragm; 3 - rod; 4 - sealing system; 5 - valve plug stem; 6 — chair; 7 - body. The usually classification criteria of the control valves are the following Control Valve Handbook, Marinoiu et al.
The modeling of the control valves represents a delicate problem because of the complexity design of the control valves, because of the hydraulic phenomena and the dependency between the elements of the control system: the process, the transducer, the controller and the control valve. From the hydraulic point of view, the control valve represents an example of hydraulic variable resistor, caused by the change of the passing section.
An overview of a control valve, together with the main associated values, is presented in figure The inherent valve characteristic of the control valve body represents a mathematical model of the control valve body that allows the determination, in standard conditions, of some inherent hydraulic characteristics of the control valve, irrespective of the hydraulic system where it will be assembled.
A control valve can not always assure the same value of the flow Q for the same value of movement h, unless there is an invariable hydraulic system. This aspect is not convenient for modeling the control valve as an automation element, because it implies a different valve for every hydraulic system.
A solution like this is not acceptable for the constructor, who should make a control valve for every given hydraulic system The inherent characteristic represents the dependency between the flow modulus of the control valve body and the control valve travel.
The flow modulus K v represents a value that was especially introduced for the hydraulic characterization of the control valves, its expression being. The way K v value was introduced through relation 35 shows that it depends only on the inherent characteristics of the control valve body, which are expressed based on its opening, so based on the movement h of the valve plug. Keeping constant the drop pressure on the valve, there is eliminated the influence of the pipe over the flow through the control valve and the dependency between the flow and the valve travel is based only on the inherent valve geometry of the valve.
The inherent valve characteristics depend on the geometric construction of the valve control body. Geometrically, the valve control body can be: a valve plug with one chair, a valve plug with two chairs, a valve plug with three ways, a valve plug especially for corner valve etc.
Consequently, the mathematical models of the inherent valve characteristics will be specific to every type of valve plug. In the following part, there will be exemplified the mathematical models of the inherent valve characteristics for the valve control body with a plug valve with one chair. For this type of valve plug, there are used two mathematical models, named linear characteristic and logarithmical characteristic, models which are defined through the following relations:.
In figure 15 there are presented the graphical dependency for the two mathematical models of the inherent characteristics of the valve control body with a plug valve with one chair Marinoiu et. Through the relation 35 he shows that the flow modulus K v has an area dimension; out of practical reasons there has been agreed to be attributed to K v a physical meaning, which would lead to a more efficient functioning.
This new meaning is based on the relation. Inherent valve characteristics types associated to the valve control body with a valve plug with one chair: 1 — fast opening; 2 — linear characteristic; 3 — equally modified percentage; 4 — logarithmical characteristic. The transformation relations are the following:.
The work characteristic of the control valve represents the dependency between the flow Q and the valve travel of the h valve plug. When defining the static work characteristic there is no longer available the restrictive condition concerning the constant pressure drop on the valve, as it was necessary for the inherent valve characteristic, but the flow rate gets values based on the hydraulic system where it is placed, the size, the type and the opening of the valve control.
From the point of view of the hydraulic system, the working characteristics can be associated to the following systems:. Due to the phenomena complexity, for the mathematical modeling of the working characteristic of the valve control body, there are introduced the following simplifying hypotheses:. The main scheme of a hydraulic system without ramifications is presented in figure Hydraulic system without branches: 1 - pump; 2 — control valve; 3 - pipe; 4 — the hydraulic resistance of the pipe.
For the modeling, the working characteristic of the valve control body, are defined by the following values:. The connection of the control valve with the hydraulic system is very tight. To be able to determine the working characteristics of the control valve there have to be solved all the elements of modeling presented in this chapter: the centrifugal pipe characteristic, the inherent valve characteristic of the control valve and the pipe characteristic.
The mathematical model of the control valve working characteristic is defined by the block scheme presented in figure The input variable is the valve travel h of the servomotor and implicit of the control valve and the value of exit is the flow rate Q which passes through the valve.
Mathematically, the model of the control valve presented in figure 17 is a nonlinear equation. The working characteristic of the control valve body, materialized by relation 40 , can be determined in two ways:. By introducing the simplifying hypothesis according to which there is considered that the flow modulus associated to the pipe does not modify, respectively ;.
By resolving numerically the model presented in figure The block scheme of the mathematical model of the body control valve. The solution has the form Marinoiu et.
In figure 18 there are presented the graphical solution of the working characteristics of the control valves for the valve plug type with inherent valve linear characteristic and inherent valve logarithmic characteristics. The solution obtained by the numerical solving of the mathematical model presented in figure 17 has been recently obtained Patrascioiu et al.
Unfortunately, the mathematical model and the software program are totally dependent on the centrifugal pump, pipe and the control valve type Patrascioiu In the following part there are presented an example of the hydraulic system model and the numerical solution obtained.
The hydraulic system contains a centrifugal pump, a pipe and a control valve. The pump characteristic has been presented in figure 3 and the mathematical model of the pump type is presented in table 1. The pipe of the hydraulic system has been presented in table 2 and the pipe mathematical model is expressed by the relation 2. The control valve of the hydraulic system is made by the Pre-Vent Company, figure 19 , the characteristics being presented in table 8 www.
The working characteristics of the control valves calculated based on the simplifying hypothesis : a valve plug with linear characteristic; b valve plug with logarithmic characteristic. The numerical results of the program are the inherent valve and the work characteristics. The inherent valve characteristic obtained by calculus confirms that the control valve belongs to the linear valve plug type. The working characteristic of the control valve is almost linear, figure The pipe drop pressure is very small, figure 5 , and for this reason the influence of the control valve into hydraulic system will be very high, see figure In this context, the inherent valve characteristic of the control valve body is approximately linear.
The control valve made by Pre-Vent Company. The work characteristic of the control valve from the studied hydraulic system. The picture presented in figure 21 is similar to the output pressure of the pump.
For this reason, the choice of a control valve with linear characteristicis wrong, the energy being taken into consideration.
The control valve drop pressure. The control valves are produced in series, in order to obtain a low price. For this goal, the control valve producers have realized the proper standards of the geometric and hydraulic properties. The control valves choice is a complex activity, composed of technical, financial and commercial elements.
Mainly, the control valves choice represents the selection of a type or a subtype industrial data based on a control valve, depending of one or many selection criteria. Technical criteria refer to the calculus of the technical parameters of the control valves. The financial elements include the investment value and the operation costs. The fluid speeds up when entering a narrower section and slows down when entering a wider segment of the pipe.
Although, the volumetric flow rate, or current, and the fluid velocity are both related to the rate at which the fluid moves, these quantities describe different fluid properties. Fluid velocity describes average motion since each fluid particle is moving in random directions, but when averaged over all the particles there is a net "ordered" motion in the direction of the fluid velocity.
Current and velocity are related to each other, since the greater current the greater the velocity. However, once the fluid is in a steady-state the current which is determined by the properties of the entire system will stay constant throughout the fluid system from one location to another. The fluid velocity, on the other hand, will change if the cross-sectional area of the pipe changes. At first, we will neglect friction and assume that internal energy stays constant, which describes non-dissipative flow.
Since speed is an indicator of kinetic energy this implies that a change in area will result in a change of translational kinetic energy-density , "density" since we describe fluids in terms of their "energy-density". We will omit in this treatment rotational motions, like the vortices that sometimes form when water goes down the drain of your bathtub, since in our model the flow is laminar.
It is only the difference in the areas between the initial and final locations that will matter. For example, if a pipe gets wider in the middle of the interval analyzed but then returns to its original width at the end, the speed of the fluid will also return to its original value, and the kinetic energy will not change between the start and the end of the pipe. In Section 5. For a dynamic fluid system we will typically ignore any height variations within a horizontal pipe since pipes are typically too narrow to result in any significant pressure changes even within a more dense liquid.
However, if we are considering a pipe which is not positioned horizontally, such as water flowing downward from a reservoir at high elevation or from a water tower, or water flowing upward to the second floor of a house from a water tank on the ground floor, changes in gravitational energy-density need to be considered.
Combining pressure and gravitational potential energy-density from Section 5. The equation above is often referred to as the Bernoulli equation. In many physical systems although friction is never completely absent, it is often negligible due to other dominating effects. For example, you can determine that a ball falling from a few meters conserves its mechanical energy to a great precision. Thus, in the case of the ball we can neglect the effects of air friction.
This is no longer true for a piece of paper where its small weight and larger surface area make air friction a much greater effect.
Likewise, in some cases friction in fluid flow is negligible compared to other transfers of energy. Whether frictional effects need to be taken into account in a flowing fluid depends on many factors, some having to do with the properties of the fluid itself, others having to do with the geometrical properties of the pipe or channel confining the fluid, and still others relate to the rate of fluid flow and the type of flow.
We treat frictional effects in fluid flow the same way we did in the energy-interaction model, by including a thermal energy term or defining an open system which loses energy as work due to friction. Although we now have a general energy conservation equation to use with many common fluid systems, we can make it much more useful by representing the rate of energy transfer to the thermal system in terms of two variables: the first is the fluid flow rate and the second is what is called the resistance of the particular section of the channel we are analyzing.
We first give a plausibility argument for why this works. Let us consider how friction comes into play in a fluid flowing through a pipe. Molecular attractions exist between the molecules of the fluid and the walls of the pipe. This will cause the molecules closest to the pipe to be essentially stationary.
So molecules a little further away from the wall of the pipe will have to slide past the molecules nearer the wall. But this sliding involves the momentary making and breaking of bonds as the molecules slide past each other, which leads to the creation of additional random molecular motion. Random molecular motion, of course, is precisely what thermal energy is. The amount of thermal energy generated by molecules sliding past one another should be less if the average fluid velocity is less.
This will occur if the rate of flow is reduced. It will also occur if the diameter of the pipe is increased, even if the overall amount of fluid flowing through the pipe remains the same. The amount of thermal energy generated should also depend on the fluid itself. Molasses molecules do not slide past each other as readily as water molecules, for example. Viscosity is the fluid property of interest here.
It turns out that it is possible to incorporate these various factors into the two parameters volumetric flow rate and resistance, which incorporates the fluid properties and the properties of the medium in which the fluid is flowing. In other words, resistance is the parameter incorporating all factors that contribute to energy dissipation, or friction, other than current.
When resistance is multiplied by current this results in the energy transferred to the thermal system per unit volume of fluid:. For most fluids, however, the resistance is independent of current for flow rates up to a certain critical value. Then it jumps up to a higher usually non-constant value as the flow becomes turbulent. However, when frictional effects are included it is now important that the energy-density terms are analyzed in the direction of current, since friction increases in the direction of motion.
In this case the steady-state fluid is flowing horizontally in a pipe with uniform area.
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