Chapter 3: Integral relations for a control volume (

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Chapter 3: Integral relations for a control volume (검사체적에 대한 적분 관계식)

■ Bernoulli equation

The Bernoulli equation is based on the application of Newton's law of motion to a fluid element on a streamline.

▶ The assumptions of the Bernoulli equation;

∙ Inviscid flow (ideal fluid, frictionless)

∙ Steady flow

∙ Along a streamline

∙ Constant density (incompressible flow)

∙ No shaft work or heat transfer

♠ Streamline

A streamline is a line that is everywhere tangent to the velocity vector at a given instant of time.

A streamline is hence an instantaneous pattern which uses for visualizing the flow pattern.

Let us consider the motion of a fluid element of length 𝑑𝑠 and cross-sectional area 𝑑𝐴 moving at a local speed V, and x is a horizontal axis and z is pointing vertically upward. The forces acting on the element are the pressure forces 𝑝𝑑𝐴 and (𝑝 + 𝑑𝑝)𝑑𝐴, and the weight 𝑤 as shown. Summing forces in the direction of motion, the s-direction, there results

𝑝𝑑𝐴 − (𝑝 + 𝑑𝑝)𝑑𝐴 − 𝜌𝑔𝑑𝑠𝑑𝐴 cos 𝜃 = 𝜌𝑑𝑠𝑑𝐴𝑎𝑠

where 𝑎_𝑠 is the acceleration of the element in the s-direction. Since the flow is steady, only convective acceleration exists

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𝑎𝑠= 𝑉𝑑𝑉 𝑑𝑠

♠ 𝑎_𝑠 is the time rate of change of the velocity of the particle, and by use of the chain rule of the differentiation, the s component of the acceleration is given by

𝑎𝑠=𝑑𝑉 𝑑𝑡 =𝜕𝑉

𝜕𝑠 𝑑𝑠 𝑑𝑡=𝜕𝑉

𝜕𝑠𝑉

Also, it is easy to see that cos 𝜃 = 𝑑𝑧 𝑑𝑠⁄ . On substituting and dividing the equation by 𝜌𝑔𝑑𝐴, we can obtain Euler’s equation:

𝑑𝑝

𝜌𝑔+ 𝑑𝑧 +𝑉

𝑔𝑑𝑉 = 0

(Note that Euler's equation is valid also for compressible flow.)

Now if we further assume that the flow is incompressible so that the density is constant, we may integrate Euler’s equation to get

𝑝

𝜌𝑔+ 𝑧 +𝑉²

2𝑔= constant

This is the Bernoulli equation, consisting of three energy heads.

♠ A head corresponds to energy per unit weight of flow and has dimensions of length.

It follows that for ideal steady flow the total energy head is constant along a streamline, but the constant may differ in different streamlines.

Applying the Bernoulli equation to any two points on the same streamline, we have

𝑝1

𝜌𝑔+ 𝑧₁+𝑉12

2𝑔= 𝑝2

𝜌𝑔+ 𝑧₂+𝑉22

2𝑔

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▶ Application of the Bernoulli equation.

Many types of devices using principles involved in the Bernoulli equation have been developed to measure fluid velocities and flowrates. There are commonly used three types of flowmeters; the orifice meter, the nozzle meter (Example 3.16), and the Venturi meter (Example 3.15).

♠ Orifice meter

∙ When measuring the flowrate of fluid eruption in the water tank (물이 담겨져 있는 수조의 분출 유속 및 유량을 측정하는 경우)

∙ Attach to the tube and calculate the flowrate using the pressure difference between the meter (관로의 도중에 부착하여 전후의 압력차를 이용하여 유량을 계산)

∙ Easy to manufacture and cheap (제작이 용이하고 가격이 저렴)

∙ Impossible for a contaminated fluid and have a large loss (이물질이 함유되어 있는 경우 사용이 불가능하고 손실수두가 크다)

단면 1에서의 속도와 압력, 단면 2의 압력을 알고, 각각의 높이를 알고 있을 때, 단면 1, 2사이에 베르누이 방정식을 적용하면

𝑝1

𝛾 +𝑣₁²

2𝑔+ 𝑧1=𝑝2

𝛾 +𝑣₂² 2𝑔+ 𝑧2

정리하면, 양쪽의 압력을 알 때

𝑣2= 1

√1 − (𝑣𝑣¹2)²

∙ √2𝑔 {(𝑝₁

𝛾 + 𝑧1) − (𝑝₂ 𝛾 + 𝑧2)}

연속방정식을 적용하고 압력과 위치에너지의 합의 차이를 h라 하면

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(𝑝₁

𝛾 + 𝑧1) − (𝑝₂

𝛾 + 𝑧2) = ℎ

단면 2에서의 속도는 다음과 같이 간단히 나타낼 수 있다.

𝑣2= √ 2𝑔ℎ 1 − (𝐴₂

𝐴1)²

만약 A1이 A2보다 월등히 크다면, 즉 1의 단면적이 2의 단면적보다 훨씬 크다면, 식은 다음과 같이 쓸 수 있다. 이 때의 식을 토리첼리의 정리라고 한다.

𝑣2= √2𝑔ℎ

♠ 토리첼리의 정리: 수조 측면 하부의 대기와 개방된 비교적 작은 구멍을 통하여 유출되는 유체(Fluid)의 속도(Velocity) 값을 계산하는 공식

실제 유동의 경우 점성과 오리피스 입출구 형상등의 오인으로 속도가 이론값보다 작아지므로, 오리피스 출구의 속도, 유동 면적, 실제유량은 수정계수를 이용하여 보정한다. 수정된다.