Mathematical expression of conservation principle of mechanical energy in macroscopic motion of fluid. 1738, the Swiss mathematician D. Bernoulli put forward this equation in the Interpretation of Force and Motion in Fluid Mechanics. It can be obtained by integrating the ideal fluid motion equation (Euler equation) along the streamline under the condition of steady flow; It can also be deduced from the first law of thermodynamics. It is a main relation in one-dimensional flow problems, and it is very important in analyzing the steady flow of incompressible fluid. It is usually used to determine the relationship between velocity and pressure in the flow process.
The form of the equation For an incompressible ideal fluid, the density does not change with pressure, and it can be obtained as follows:
2PuZg+, constant, 2 rho
Where z is the height from the datum plane; P is static pressure; U is the fluid velocity; ρ is the fluid density; G is the acceleration of gravity. In the equation
M/kg, in which the left three terms, called potential energy in turn, are the mechanical energy per unit mass of fluid, and the unit is n..
Term, hydrostatic energy term and kinetic energy term. The equation shows that the three kinds of energy can be converted to each other, but the sum remains the same. When the fluid flows in a horizontal pipe, z is a constant, and the above formula can be simplified as:
2uP, constant, 2ρ
This formula expresses the relationship between flow velocity and pressure: where the flow velocity is high, the pressure is low, and where the flow velocity is low, the pressure is high.
For the fluid with unit weight, based on 1 and 2 sections of the pipeline, the form of the equation becomes:
22PuPu 1 122ZZ2，，，，， 1 G2 gg2gψ
In the formula, each term is the energy per unit weight of the fluid and has the dimension of length. The three terms are called position head, static head and dynamic head (velocity head) in turn.
For an ideal compressible fluid, the density varies with pressure. If this change is a reversible isothermal process, the equation can be written as follows:
22uuPP 12 12 gZ，，gZ，，ln 1222ρP 1 1
If it is a reversible adiabatic process, the equation can be written as:
22uuPP 12 12 gZ，，gZ，，ln 1222ρP 1 1
In Cv formula, it is the ratio of constant pressure specific heat capacity to constant volume specific heat capacity, that is, specific heat capacity ratio, also known as adiabatic index. cellulose propionate
For viscous fluids, there is a velocity distribution on the flow section. If the kinetic energy term is expressed by the average speed, it must be multiplied by the power u.
Dο energy correction coefficient. In addition, it is necessary to consider the flow resistance caused by viscosity, that is, the loss of mechanical energy hf per unit mass of fluid. If in the process of fluid flow, the unit mass of fluid receives the work w done by the fluid conveying machinery. Under these conditions, if we take two segments 1 and 2 in the uniform flow segment as the benchmark, the equation can be expanded as follows:
When the laminar flow of fluid in a circular tube =2, this value can be calculated from the velocity distribution; When turbulence occurs, 1.06.,,,
Application of Equation The principle of mutual conversion of potential energy, kinetic energy and static pressure energy expounded by Bernoulli equation can be used to analyze and calculate some practical problems, such as:
Calculate the flow rate of the fluid flowing out of the orifice. The container is full of liquid, and the liquid level remains unchanged. A small hole is opened on the wall of the container at H position under the liquid surface, and the liquid flows out from the small hole under the action of gravity. According to Bernoulli equation, it can be calculated that the average velocity of liquid flowing out of the orifice is:
u，Cd2gh
In formula C, d is the pore flow coefficient, which is determined by experiments and is about 0.6 1, 0.62; G is the acceleration of gravity. According to the above speed and the known orifice area, the flow through the orifice can be calculated; Or according to this relationship, calculate and determine the liquid level height that must be maintained to reach a certain flow. If the gas flows out of the small hole in the wall of the container under a certain pressure difference, it can be regarded as an incompressible fluid under the condition of not too high velocity, and its flow rate can also be estimated by Bernoulli equation.
u？ Pitot tube is equipped with uniform airflow and flows around the object at a constant speed. After the airflow is blocked, stop at the front edge of the object (a), 0.
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Form a stagnation point (figure 1 stagnation point), and the pressure at this point is called stagnation pressure. If the O pressure at the undisturbed point is 0, it can be obtained from the Berppa o Nuri equation.
U measure the difference between and, and you can calculate the flow. The velocimeter designed according to this principle is called velocimeter, also called pitot tube. Bippao 0
Pitot tube (Pitot tube structure in Figure 2) consists of a double-layer sleeve with a round head, and the center of the round head is provided with a small hole connected with the inner sleeve, and the inner sleeve is connected with one end of a pressure gauge to measure the stagnation pressure; At a certain distance from the side surface of the outer sleeve, the pA is uniformly opened in the circumferential direction.
A row of static pressure holes perpendicular to the pipe wall, and the outer sleeve is connected with the other end of the pressure gauge to measure the pressure. According to the measured pressure difference p0
H, you can calculate the velocity of the measuring point.
Venturi tube, also known as Venturi tube (Figure 3), is a kind of pipe that contracts first and then expands gradually. Because the cross-sectional area has changed, the flow rate has changed, and according to Bernoulli equation, the pressure has also changed. The flow can be calculated by measuring the pressure difference between the pipe and the throat. The venturi used to measure the flow is called a venturi flowmeter. Due to the high-speed airflow formed at the throat of Venturi tube, negative pressure will be generated to suck the liquid, so that the gas and liquid will be in close contact, which can be used to complete gas washing, cooling, absorption, reaction and other operations. The venturi used for this operation is called a venturi scrubber.
1. Bernoulli
1700 65438+1October 29th, Bernoulli was born in Switzerland (he is not only a physicist, but also a mathematician) (in the late 1940s, he published his famous book Fluid Mechanics, in which he solved the problem of fluid flow with the law of conservation of energy. He analyzed the relationship between pressure and velocity when fluid flows, and obtained the equation based on him later. In the book, Bernoulli also clearly describes the theory of molecular dynamics, and thinks that the pressure of gas acting on the wall can be explained by the rapid back and forth movement of a large number of molecules. He also published a paper on tidal and chord vibration of seawater, and made outstanding contributions in the fields of calculus, differential equations, probability theory and so on. During the period of 1725 and 1749, Bernoulli won the annual prize of French Academy of Sciences for ten times.
Bernoulli's experiment shows that when the ideal fluid flows stably, the pressure is low where the speed is high, and strong (but not inversely proportional) where the speed is low. Its mathematical expression is as follows
, p+ρv,,, ρgh, constant
This is the famous Bernoulli equation.
2. Solve practical problems with Bernoulli equation.
(1) Determine the pressure at the depth h under the static liquid surface.
As shown on the right, in a container filled with liquid, take the point A on the liquid surface and the depth below the liquid surface.
H point B, with the horizontal plane of point B as the zero (potential energy) datum, then
h、、、、、、、、、、 1BA0
Because the liquid is static, v,, and are substituted into Bernoulli equation to get 12.
p，，，ρgh，，，ρgh，A0
(2) Find the velocity of the liquid flowing out of the small hole.
There is a small hole on the wall of the container, the depth of which is h, which is lower than the liquid level. The liquid flows out from the small hole and takes away the liquid.
The point A on the surface and the point B at the small hole are studied. Because the cross section of the container is much larger than that of the small hole, so
Because the water level in the container drops slowly, the velocity of liquid particles at point A can be ignored, that is, v, a.
If the height of point B is zero, then points A and B are in contact with the atmosphere, so B.
(atmospheric pressure) and substitute it into Bernoulli equation to get p, AB0.
122ghp, ρgh,,, ρv is V, 00BB2.
(3) measuring the flow rate of the fluid
When measuring the velocity of fluid in a pipeline, the instrument shown in the following figure can be used, because it is often used to measure the velocity of air flow, so
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call out
Make an airflow speedometer and put the pitot tube A (pitot tube is an elbow with one end closed, and the closed end A is smooth and slightly pointed, leaning against it.
There are many small holes on the side near the closed end) and a tube B (dynamic pressure tube) with the nozzle facing the airflow are connected to the U-tube pressure gauge.
In fact, according to the height difference of the liquid column on both sides of the U-shaped tube, the flow rate of gas can be obtained (
Let the speed of stable gas flow be V, the density of gas be ρ, and the liquid in the pressure gauge be 0.
The density is ρ, the gas pressure at the small hole of pipe A is P, and the gas pressure of pipe B is 0A.
P, because the gas in the pipeline is blocked by the fluid in the pipeline, and its flow is equal to 0, because the pipeline ABB
According to Bernoulli equation, at the same height as the port of pipe B and on the same gas streamline.
get
2p, ρv,,, p,, B2, so p, p, ρv,, (b
According to the height difference H between the two sides of the U-shaped tube, the pressure difference of gas in the two tubes can be calculated as follows
p，p，ρgh B，0
It consists of v2, gh/ 0 of the above categories.
Therefore, the speed of air flow can be measured by measuring h (
(1) air intake of liquid flow and air flow
As shown in the following figure, if a small hole A is made in the neck of the horizontal tube and a thin tube is connected to the container in container B, the flowing liquid will not flow out and the liquid in container B can be sucked up. In order to study this principle, the following deduction is made:
Assuming that the upper left container E is large, the liquid level does not drop significantly when the fluid flows. The height difference between the liquid level and the liquid outlet hole is h, s, and s respectively represents the cross-sectional area of the orifice A and the liquid outlet hole F on the horizontal pipe, ρ represents the density of the liquid, and f is the liquid.
For the ideal fluid, taking the point C on the liquid level in the container E, the water in the small hole A and the water outlet hole F in the horizontal pipe as the research objects, according to Bernoulli equation, the following results are obtained:
1 122p，ρgh，p，ρv，，，ρv C，FAF22
And because P, P, and are substituted into the above formula, CF02V and ρ GH F are obtained.
122,, Vice President,,, (,) A0FA2
According to vSFA, when the fluid flows stably in the horizontal pipeline, the flow of each part of the pipeline, ρvSt, remains unchanged, including: ρ vSAF.
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S, S (then, a) is obtained by synthesizing the above formulas.
2S 1Fp，p，ρgh(，，)，A022SA
That is, there is a certain degree of vacuum at the hole C, so the liquid at the place B can be sucked in. This phenomenon is called cavitation (
Not only does the liquid flow have a suction function, but the air flow also has a suction function, and the law followed is the same. The inspiratory effect is widely used. The water pump in the chemical laboratory, the carburetor of the internal combustion engine and the water injector used to add water to the steam boiler are all made according to this principle (
References:
Application of Bernoulli Equation in Middle School Physics Teaching in 2007
Brief introduction to the principle of fluid chemical engineering
Solid fluidization technology is an important branch of chemical engineering. Fluidized bed has high heat and mass transfer efficiency and the ability to handle a large number of particles, so it is widely used in chemical industry, petroleum processing, energy, environmental protection, food processing, pharmaceutical production and other fields. Scientific research closely related to industrial practice is also extremely active, and new scientific research achievements and theories are constantly emerging. With the further development of basic scientific research and national economy, fluidization technology will be applied in more fields.
This book is the first monograph on solid fluidization in China, which was completed by 16 experts and famous scholars at home and abroad for several years. As a monograph, the book covers almost all the important contents of fluidization. The book is divided into 1 1 chapter: chapter 1 introduces fluidization phenomenon and its development history; Chapter 2 introduces the related basic knowledge; In chapter 3, chapter 4 and chapter 5, the flow laws of gas-solid dense fluidized bed, circulating bed and parallel gravity field fluidized bed are introduced in detail. Chapters 6 and 7 discuss the heat transfer of fluidized bed and the model and amplification of reactor; Chapter 8 describes the basic characteristics of spouted bed; Chapter 9 gives many examples of industrial application of fluidized bed; 10 chapter specifically introduces the experimental technology and test method of fluidized bed; Chapter 1 1 introduces the recent development of liquid-solid dispersed fluidization and gas-liquid-solid three-phase fluidized bed.
This book can be used as a reference for scholars, researchers, engineers and technicians, operation and management personnel engaged in fluidization work, and can also be used as a teaching material and teaching reference book for chemical engineering, petroleum, heat energy and other related majors in colleges and universities.
This section introduces the basic concept of fluidization, the main characteristics, advantages and disadvantages of operation of fluidized bed with flow resistance.
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Simply put ~ solid fluidization is the fluidization of solid matter. The fluid flows at a certain speed.
When passing through a bed composed of solid particles, a large number of solid particles can be suspended in a flowing fluid.
Particles roll up and down under the action of fluid, just like liquid. This state is the flow pattern.
Change.
Fluidization is an important technology in chemical industry and many other industries, such as energy and metallurgy.
A widely used industrial technology. It is mainly used for strengthening heat transfer and transmission in chemical industry.
Mass ~ (2+) can also realize gas-solid reaction, physical treatment and even particle transport.