kinetic energy of water flowing through a pipe formula

A_1 = \pi \times ( 0.065 \ \rm m )^2 \\ Find the discharge through the pipe if the velocity of water flowing through the pipe section 1 is 5 m/s. The stages mentioned above now repeat. Find the piezometric head of the water at a cross-section which is 5 m above the datum line. The pressure at A is 700 kPa and the pressure at B is 125 kPa. The force does not come from the pressure there, however; it comes from your hand taking momentum away from the fluid. It is estimated that due to friction in the piping there is an energy loss of 4 N.m/N of water flowing: (a) Calculate the power delivered to the fluid motor by the water << At the same time, other factors such as change in velocity and elevation also lead to energy dissipation. /Name /Im0 Q Although the picture here shows nice streamline flow as the air moves left relative to the ball, in reality there is some turbulence. Consider a geyser that shoots water 25 m into the air. As a rough estimation one can say the total energy of the water flowing out through . .IO��""�MR���lԁ�>oo��C�����?p���@I���ȣ�JK��I a. /Subtype /Image To throw a curveball, the rotation of the ball should be around a vertical axis. This causes air that travels below the ball in the picture to move up and fill the space left by the ball as it moves by, which reduces drag on the ball. 2 0 obj /Type /Page © copyright 2003-2021 Study.com. Example: Determine the kinetic energy of 7 lbm of steam flowing through a pipe at a velocity of 100 ft/sec. {/eq}. The gauge pressure of water at the level of the opening of the intake pipe is {eq}P = d \rho h \\ Mechanical energy of a flowing fluid per unit mass : Flow energy + kinetic energy + potential energy. Which pressure is greater? Is the pressure higher or lower in the narrow section, where the velocity increases? If a pipe is attached at the bottom water flows along this pipe out of the tank to a level . The side where the air speed is higher has lower pressure, so the ball is deflected toward that side. Another way to do it is to apply Bernoulli's equation, which amounts to the same thing as conservation of energy. Discover what gadget is used to measure air pressure along and how it works . The Bernoulli equation (energy equation) calculation does not check for unreasonable inputs such as negative density, negative velocity, or pressure less than a complete vacuum. For turbulent flow, the speed and or the direction of the flow varies. q{�����ࠒ �� r���;���׿����?���2\]]Gw�u�]|>�V�}Bx��,ˌ>[�ޚw����� ����������w˖-o����?Θ��_~���裏F�A===�Bq�3> Fluids can flow steadily, or be turbulent. There are basically two ways to make fluid flow through a pipe. /Contents 4 0 R In a flowing fluid, potential energy may in turn be subdivided into energy due to position or elevation above a given datum, and energy due to pressure in the fluid. A_1 \approx 1.327 \times 10^{-2 } \ \rm { m^2 } The second way is to make the pressure at one end of the pipe larger than the pressure at the other end. The circular nozzle is 6.5 cm in radius. Let's take point 2 to be 25 m above ground, which is 60 m above the chamber where the pressurized water is. section that is in contact with the flowing fluid. The solution of pipe flow problems requires the applications of two principles, the law of conservation of mass (continuity equation) and the law of conservation of energy (Bernoulli's equation) 1.1 CONSERVATION OF MASS When a fluid flows at a constant rate in a pipe or duct, the mass flow rate must be the same at all points along the length. ouN \/���[�n okk�������Y�d�?���~��翺Y�CCC����ڮ� Therefore the pressure difference between the water at the intake pipe opening and the water streaming out of the nozzle is {eq}\Delta P = P_{2a} - P_a \\ Bernoulli was a Swiss mathematician who studied energy conservation in the 1700s. The intake pipe of radius {eq}r_2 = 15 \ \rm { cm } As the water gushing out through the nozzle goes to a height h, the velocity of the water at the nozzle is {eq}v_1 = \sqrt { 2 g h } \\ We will see that it depends on both the dimensions of the container and the speed of the fluid. x��\TG��IS��K�K�ߥ\�]��]�1�ihb�g���.��V� Torricelli's Theorem: Tank Experiment, Formula and Examples. Water flows horizontally through a garden hose of radius 1.0 cm at a speed of 1.4 m/s. << /ColorSpace 10 0 R All other trademarks and copyrights are the property of their respective owners. Bernoulli's theorem is based on the principle of conservation of energy for a flowing fluid. Acceleration due to gravity {eq}g = 9.8 \ \rm { m/s^2 } The water pump supplies kinetic energy . When fluids flow through pipes, energy losses inevitably occur. As air flows over the ball, the seams of the ball cause the air to slow down a little on one side and speed up a little on the other. In this lesson, we will investigate a property of fluid flow called the Reynolds number. Explore the definition, application, and examples of Bernoulli's principle and learn about streamlines. Pressure vs. speed. When the velocity profile in a pipe is uniformly distributed, then α = 1. From eq. Learn how to apply Bernoulli's principle equation by various examples. What is the velocity of the water as it enters the intake pipe? Torricelli's Theorem describes the relationship between the height of fluid in a container and the velocity of fluid exiting through a hole in that container. For our first look at the equation, consider a fluid flowing through a horizontal pipe. The speed of flow of water in the pool {eq}v = 0 We have . In this lesson, you'll explore the relationship between angular and linear velocity. %PDF-1.4 The pipe is horizontal, so both points are at the same height. The kinetic energy of water is a result of the speed or flow rate of the water. Pascal's principle states that a change in pressure at any point in an enclosed fluid is transmitted equally throughout that fluid. Consider an incompressible fluid flowing through a pipe that has a varying diameter and height, as shown in . [ /PDF /Text /ImageC ] A portion of energy is lost to overcome the resistance to the flow. The turbines, boilers and pumps in large-scale power generation plants are open systems. {/eq}, If {eq}v_2 The buoyant force is the force exerted by a fluid to oppose the weight of the object placed in it. Fluids in Physics: Definition and Characteristics. Passing a certain volume of liquid through a pipe requires certain amount of energy. A_2 = \pi \times ( 0.15 \ \rm m )^2 \\ The following pieces of information are given in the question. /Resources << The Doppler effect causes wave frequencies to change when the source of the waves is moving. There are three different types of energy within the flowing fluid - flow energy (pressure head), kinetic energy, and potential energy. >> Kinetic energy (KE) is the energy of motion. According to Bernoulli's theorem, the total energy the sum of the pressure energy, kinetic energy, and potential . "�[}��[�Z|䟥�?f����ٕ���p���Q��Ki�)a��"���W�t!8.6����Vg n�?��_���e����srr���FD8z�������}��u��644���k������=P�������_{�ȡC���;::��c,����:h�j��Rʻ� �^������w�2e�M����O���{۶m ���"5`gg�u������o�eji��I��ڻi���K�����̠z�U��ԝ���7m�����gefj�$8����Q�ފąkx�M�}�>� �b"���rss��;� ���c�d����?���SO1_y�W�O�l��-۶����4~��H�:$5y�244��V���M��+������>�=���O�lm��\,���K]naf|����W��* `�(,,$FF"���i��Cz�R;PV&���%$ v_1 = 17.7087549 \ \rm { m/s } \\ \Delta P = P_2 + P_a - P_a /Pages 2 0 R This can get very complicated, so we'll focus on one simple case, but we should briefly mention the different categories of fluid flow. The pressure of water streaming out of the nozzle is the same as that of atmospheric pressure. Learn how a heavy object floats, and understand the definition, types, equation, and example of buoyancy. P = 7350 \ \rm { P a } The pressure, speed, and height (y) at two points in a steady-flowing, non-viscous, incompressible fluid are related by the equation: Some of these terms probably look familiar...the second term on each side looks something like kinetic energy, and the third term looks a lot like gravitational potential energy. The First Law for open systems. Internal Energy of a System: Definition & Measurement. /ProcSet 6 0 R >> There are basically two causes that make a fluid flow through a pipe. {/eq}, Therefore the gauge pressure at the opening of the intake pipe is {eq}P_2 = P - \dfrac { 1 } { 2 } \rho v_2 ^2 \\ When flow is turbulent, the velocity profile is plug-like and α ≈ 1.05. Both energy and work are measured in Newton-meter (or pounds-foot in English). {/eq} may be used for the calculations. q v_1 \approx 18 \ \rm { m/s } The diameters of a pipe at the sections 1 and 2 are 10 cm and 15 cm respectively. Let Q is the discharge through the channel, b is the channel width, h is the depth of flow and q is the discharge per unit . Pressure drop in flow is proportional to viscosity. Kinetic energy and potential energy are the two commonly recognized forms of energy. is the speed of water flow at the intake pipe, the equation . >> The density of the streamlines increases as the velocity increases. ?�o���� ��^�p�­�5 �륣���ѣ_}��믿>u�ԉ����\ �L���������?���g�yF�|�� We often talk about individual types of energy, but how do you measure total energy? {/eq} is the speed of water flow at the intake pipe, the equation of continuity can be expressed as {eq}A_1 v_1 = A_2 v_2 /XObject << /Im0 8 0 R >> What is the kinetic energy of the water in (ft)(lb_{f})/lb_{m}? >> The mass flow rate is simply the rate at which mass flows past a given point, so it's the total mass flowing past divided by the time interval. Mass flow rate is measured in [ kg s k g s ]. To determine the pressure 35 m below ground, which forces the water up, apply Bernoulli's equation, with point 1 being 35 m below ground, and point 2 being either at ground level, or 25 m above ground. {/eq}, Area of cross-section of the inlet pipe {eq}A_2 = \pi r_2 ^2 \\ In the Bernoulli equation, the energies are represented as pressures. {/eq}, Absolute water pressure at the intake pipe is {eq}P_{2a} = P_2 + P_a \\ Summarising 2 Initial kinetic energy =0 Initial potential energy =mgz1 Final kinetic . Example: Determine the kinetic energy of 7 lbm of steam flowing through a pipe at a velocity of 100 ft/sec. This applies to a liquid that occupies part of a conduit, as in sewer lines carrying waste-water, or a creek or river. b. In the Bernoulli equation, the energies are represented as pressures. endobj In classical mechanics, kinetic energy (KE) is equal to half of an object's mass (1/2*m) multiplied by the velocity squared. A Liquid Flowing Through a Horizontal Pipe of Non-Uniform Area of Cross-Section. Pressure vs. speed. Bernoulli's equation has some surprising implications. In steady flow, the fluid passing a given point maintains a steady velocity. {/eq}, density of water {eq}\rho = 1000 \ \rm { kg/m^3 } The potential energy of the water in the rest of the pipe is the same as the potential energy of the water that used to be in the rest of the pipe before the movement. Looked at in that way, the equation makes sense: the difference in pressure does work, which can be used to change the kinetic energy and/or the potential energy of the fluid. - Definition & Formula. If a pipe is attached at the bottom water flows along this pipe out of the tank to a level . In a flowing fluid, potential energy may in turn be subdivided into energy due to position or elevation above a given datum, and energy due to pressure in the fluid. In the pumping process, no water is lost or there is no source and sink of water between the inlet point and the outlet point. {/eq} be the pressure at the opening of the intake pipe. From Equation (4) we get: And from Equation (6): Combining these two equations we get: So if we divide the kinetic energy Equation (23) by (A.S) we get it in terms of pressure: The A.S terms cancel out to give the kinetic energy in terms of a pressure (P k): << For example, if a an object with a mass of 10 kg (m = 10 kg) is moving at a velocity of 5 meters per second (v = 5 m/s), the kinetic energy is equal to 125 Joules, or (1/2 * 10 kg) * 5 m/s 2 . HE@�H����{YzgA�؅]�ea�����b��}?����Λ73�0�7�)� e}}}PP��Çɿ�����V� ��������g�ڵ3gμ���u8����{n޼y;v�8w�\III__߭�, �I ? Let's do it that way, just to convince ourselves that the methods are the same. The Doppler Effect: Formula & Calculation. For pipes there is Darcy-Weisbach, for fixtures like elbows or valves there is the equivalent length. Solving for velocity gives v = 22.1 m/s. Learn the mass flow rate equation and volume flow rate equation. /BitsPerComponent 8 Your first inclination might be to say that where the velocity is greatest, the pressure is greatest, because if you stuck your hand in the flow where it's going fastest you'd feel a big force. where: is the kinetic energy of the fluid in joules per cubic meter (J); is the volume of the fluid (m 3); is the density of the fluid in kilograms per cubic meter . Kinetic energy (KE) is the energy of motion. Fluid dynamics is the study of how fluids behave when they're in motion. /Length 5 0 R Using English system units, it is defined by Equation 1-12. Water is flowing through a 2-in.-diameter pipe with a velocity of 3 ft/s. {/eq}. After completing this lab, you will be able to explain how lenses work, explain the difference between convex and concave lenses, draw ray diagrams to analyze the image produced by a lens, and apply that understanding to eyeglasses. 3 0 obj endstream {/eq}, and atmospheric pressure {eq}P_a = 1.013 \times 10^5 \ \rm { P a } This lesson looks at the definition of static pressure as well as how to compute the pressure. What is the flow rate in gallons per minute? The absolute pressure of water in a pipe is 1.02 x 10 5 N/m 2 when the water is flowing at 1.7 m/s. Kinetic Energy is the energy an object has owing to its motion. a. Solution: Using Equation 1-12. Looked at in that way, the equation makes sense: the difference in pressure does work, which can be used to change the kinetic energy and/or the potential energy of the fluid. As energy cannot be created, it follows that the kinetic energy in molecule 3 is formed from that pressure energy held in molecule 2 immediately before the plug was removed from the hole. There are three different types of energy within the flowing fluid - flow energy (pressure head), kinetic energy, and potential energy. For example, if a an object with a mass of 10 kg (m = 10 kg) is moving at a velocity of 5 meters per second (v = 5 m/s), the kinetic energy is equal to 125 Joules, or (1/2 * 10 kg) * 5 m/s 2 . The sketch below shows a jet pump (called an... Bernoulli's Equation Example | What is Bernoulli's Equation Formula? You'll start with a refresher on how to find angular velocity, and then learn how to convert angular velocity into linear velocity. /Parent 2 0 R Kinetic Energy. Kinetic Energy of Water. The relationship for the kinetic energy per unit volume of water is thus proportional to its velocity and can be expressed as:. The intake pipe is 75 cm below the surface of the pool and 15 cm in radius. v_1 = \sqrt { 2 \times 9.8 \ \rm { m/s^2 } \times 16 \ \rm m } \\ Equation of continuity can be applied to the water flow through the inlet and outlet of the pump. Our experts can answer your tough homework and study questions. An open system allows mass flows across the system boundary. Flow rate at the wall of the pipe is actually 0 and flow rate in the center of the pipe is higher than average. {/eq} is kept at a depth {eq}d = 0.75 \ \rm { m } 7.3 Energy Equation: Pipe Flow Kinetic Energy Correction Factor In most cases, α takes on a value of 1 or 2. The loss term h L accounts for all minor (valves, elbows, etc.) Know how matter exerts pressure and learn its definition. Looked at in that way, the equation makes sense: the difference in pressure does work, which can be used to change the kinetic energy and/or the potential energy of the fluid. 1 0 obj How fast is the water traveling when it emerges from the ground? For our first look at the equation, consider a fluid flowing through a horizontal pipe. P = 0.75 \ \rm m \times 1000 \ \rm { kg/ m^3 } \times 9.8 \ \rm { m/s ^2 } \\ /Filter [ /FlateDecode ] In classical mechanics, kinetic energy (KE) is equal to half of an object's mass (1/2*m) multiplied by the velocity squared. The . 6 0 obj Bernoulli's Principle: Definition and Examples. {/eq}. When flow is turbulent, the velocity profile is plug-like and α ≈ 1.05. And from Bernoulli's theorem, we know that the sum of potential energy, kinetic energy and pressure energy remains constant. Static pressure is the pressure that a fluid exerts when it is not moving. Pressure Units - How is Air Pressure Measured? The "Bernoulli equation without friction" (as derived and shown in Chapter 4, page 87 of Unit Operations of Chemical Engineering by Warren L. McCabe 7th Edition) is possibly a useful energy balance equation for a unit mass flowing along a streamline through a tube. v_2 \approx 3.3 \ \rm { m/s } What is the kinetic energy of the water in (ft)(lb_{f})/lb_{m}? In this lesson, you'll learn how to quantify the amount of fluid flowing through a pipe, channel, or other container. If the equation was multiplied through by the volume, the density could be replaced by mass, and the pressure could be replaced by force x distance, which is work. /Kids [ 3 0 R ] In streamline flow, the product of cross section area and velocity remains constant (equation of continuity). {/eq}. Find out the absolute pressure if the velocity of water is changed into 23 m/s. Kinetic energy and potential energy are the two commonly recognized forms of energy. After watching this video, you will be able to explain what simple harmonic motion is, describe the energy interplay involved in various situations, and use energy equations to solve simple harmonic motion problems. >> We can take the velocity to be zero at both points (the acceleration occurs as the water rises up to ground level, coming from the difference between the chamber pressure and atmospheric pressure). {/eq} into the air. The kinetic energy is now z2 m u2 1 2 2 mu2 and the potential energy mgz. A water pump is designed to send a stream of water through a nozzle of radius {eq}r_1 = 6.5 \ \rm { cm } Bernoulli's principle states that as a fluid's speed increases, the fluid's pressure decreases. Water at 10°C is flowing at a rate of 115 L/min through the fluid motor shown in the figure. Let's say the ball is thrown so it spins. << Summarising 2 Initial kinetic energy =0 Initial potential energy =mgz1 Final kinetic . Solution: Using Equation 1-12. According to the equation of continuity, the volume of water flowing in through the inlet pipe per second will be equal to the volume of water flowing out through the outlet pipe per second. Density & Buoyancy of Objects: Physics Lab. How to Convert Angular Velocity to Linear Velocity. Blood flows through the body in a specific way, and in general, we want that to continue happening. /CropBox [0 0 1200 4161] A water pump is a mechanical device for drawing water from a reservoir through the inlet pipe and supply at the desired location through the outlet pipe. For our first look at the equation, consider a fluid flowing through a horizontal pipe. Kinetic Energy is the energy an object has owing to its motion. In other words: bouncing back kinetic energy isn't going to help. Air and liquids move, which is called flow. Kinetic Energy of Water. In a nutshell, the theory named after him states that the number of all energies - static, potential, and kinetic - in a fluid flowing through a pipe stays constant within the pipe. At some point in the water jet immediately after passing through the hole, molecule 3 is to be found in the jet and will have a certain velocity and therefore a certain kinetic energy. P_{2a} = 103138.8 \ \rm { P a } \\ Now, consider a rectangular channel with a steady but non-uniform flow. endobj The air does exert a force down on the ball in the figure above, so the ball must exert an upward force on the air. v_2 = 3.322662388 \ \rm { m/s } \\ Let {eq}P_2 The pipe is narrower at one spot than along the rest of the pipe. \Delta P = 1838.8 \ \rm { Pa } \\ Explore the interference pattern and equations that explain the observed phenomena with light passing through double-slits. By applying the continuity equation, the velocity of the fluid is greater in the narrow section. /Length 9 0 R change in potential energy = (mass of water in V)*g*(change in height) = density*V*g*(h 2 - h 1) = ρVg(h 2 - h 1). Learn the equation of continuity and its derivation. << This is what Bernoulli's equation does, relating the pressure, velocity, and height of a fluid at one point to the same parameters at a second point. Whenever the fluid is flowing, there will be an energy that causes the fluid to flow. Introduction. Kinetic Energy. In this lesson, learn about the Doppler effect and how to calculate the frequency of sound produced by a moving source. For Bernoulli applications, please see our Bernoulli Calculator with Applications. 8 0 obj A short quiz will follow. One way is to find the internal energy of a system. According to the equation of continuity of fluid flow, if a liquid flows in streamlined motion through a tube of non-uniform cross-section, then the product of the area of cross-section and the velocity of flow is the same at every point in the tube. We can measure the potential energy from ground level, so the potential energy term goes away on the left side, and the kinetic energy term is zero on the right hand side. The potential energy of the water in the rest of the pipe is the same as the potential energy of the water that used to be in the rest of the pipe before the movement. The pressure on the right-hand side is atmospheric pressure, and if we measure heights from the level of the chamber, the height on the left side is zero, and on the right side is 60 m. Bernoulli's equation can be simplified in this case to: The kinetic energy term on the right is larger than the kinetic energy term on the left, so for the equation to balance the pressure on the right must be smaller than the pressure on the left. A water pump is designed to send a stream of water from a nozzle at the surface of a pool of water 16 m into the air. /MediaBox [0 0 1200 4161] It is this pressure difference, in fact, that causes the fluid to flow faster at the place where the pipe narrows. /Type /Pages The kinetic energy is now z2 m u2 1 2 2 mu2 and the potential energy mgz. In steady flow, the motion can be represented with streamlines showing the direction the water flows in different areas. \Delta P = 1.8 \times 10^3 \ \rm { Pa } The kinetic energy of water is a result of the speed or flow rate of the water. We may use Bernoulli's equation to connect the total energy of the water at a point just at the opening of the intake pipe and another point inside the pool at the same horizontal level. Feeling pressured to learn about Pascal's principle? What is Static Pressure? As a rough estimation one can say the total energy of the water flowing out through the outlet will be equal to the sum of the total energy of the water drawn through the inlet and the kinetic energy supplied by the pump. Explore the definition and properties of fluids including compressibility, density, pressure, buoyancy and viscosity. From Equation (4) we get: And from Equation (6): Combining these two equations we get: So if we divide the kinetic energy Equation (23) by (A.S) we get it in terms of pressure: The A.S terms cancel out to give the kinetic energy in terms of a pressure (P k): Energy or pressure difference must exist to cause liquid to flow. Suppose that the pipe in question carries water, A1 = 2.0A2, and the fluid heights in the vertical tubes are h1 = . ��R[���2f���!R'gECC�����N�Ԥ�����vw˓S������}g. See examples and learn more. Simply put, what goes in - must come out. The equation of continuity can be reduced to: Generally, the density stays constant and then it's simply the flow rate (Av) that is constant. Hydrostatic Pressure: Definition, Equation, and Calculations. endobj A double-slit diffraction is the observed effect where light passing through two slits creates beams that interact with each other. Answer (1 of 3): How can I calculate the velocity of water flowing through a pipe of two circular cross-section area A¹ and A² lying horizontally? (11.3) Specific energy of a flowing liquid, Where, EP is the potential energy of flow (h) and Ek is the kinetic energy of flow (v 2 /2g). We will examine how, where, and why this occurs in the body. Fluid flow can be rotational or irrotational. 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When flow is laminar, the velocity distribution is parabolic and α = 2. Understand how to calculate the mass flow rate with volume flow rate. The relationship for the kinetic energy per unit volume of water is thus proportional to its velocity and can be expressed as:. /Type /Catalog P_2 = 7350 \ \rm { P a} - \dfrac { 1 } { 2 } \times 1000 \ \rm { kg/ m^3 } \times ( 3.32 \ \rm { m/s })^2 \\ Pascal's Principle happens when a change in pressure forced on an enclosed liquid is transmitted to all of the fluid and the walls of the container it is in. Then you came to the right place. We have . Let's learn what it means and how it is measured. P_{2a} \approx 1.03 \times 10^5 \ \rm { Pa } �@oGM}�!����q�C�맭��G�L�Ή8]��ϟ=��9�cMN�ybro�r �dQSSC��fH__��>>>7n|�7x�M'8eʔ�{����,ŗ�Z One way is to tilt the pipe so the flow is downhill, in which case gravitational kinetic energy is transformed to kinetic energy. A pressure difference is like a net force, producing acceleration of the fluid. For most of the rest of the chapter, we'll focus on irrotational, incompressible, steady streamline non-viscous flow. To figure out how fast the water is moving when it comes out of the ground, we could simply use conservation of energy, and set the potential energy of the water 25 m high equal to the kinetic energy the water has when it comes out of the ground. endobj P_2 = 1838.8 \ \rm { P a } Mechanical energy change: • The mechanical energy of a fluid does not change during flow if its pressure, density, velocity, and elevation remain constant. /Count 1 Learn more about Torricelli's Theorem, the formula for what is often referred to as the tank experiment, and practical examples, including a spouting-can experiment and a leaking dam. Learn about the various units of pressure and their abbreviations such as atm, millimeters of mercury and Psi. Bernoulli's equation has some surprising implications. On the one hand, this is due to friction that occurs between the pipe wall and the fluid (wall friction).On the other hand, frictional effects also occur within the fluid due to the viscosity of the fluid (internal friction).The faster the fluid flows, the greater the internal friction effect (see also the article on . 7 0 obj If the water originates in a chamber 35 m below the ground, what is the pressure there? The jet engine of an aircraft is an open system. After completing this lab, you will be able to explain what buoyancy is, how to determine the size of the buoyant force, and use the principles of buoyancy to calculate the density of an object. >> Using English system units, it is defined by Equation 1-12. If a gas flows through a conduit, the entire perimeter is "wetted." Using the above definition, we arrive at the following results for the equivalent diameter for two