Short answers on fluid mechanics

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A fluid is a substance that undergoes continuous deformation under any slight shear force. Continuum hypothesis: The continuum hypothesis regards a fluid region as being composed of continuous fluid particles that fill space without any gaps, with their physical properties and motion parameters being continuously distributed throughout space. Thus, the unevenness, discreteness, and irregularity of microscopic motion achieve a harmonious unity with the uniformity, continuity, and regularity of macroscopic motion. Purpose of the continuum hypothesis: By treating a fluid with microscopic discontinuities as a continuous medium, its physical quantities become continuously distributed throughout the flow field. This not only allows for the use of mathematics, a powerful tool, in theoretical analysis but also makes experimental studies possible. Density of water: 1000 kg/m3; density of mercury: 13600 kg/m3. Viscosity is significantly affected by temperature: as temperature rises, the viscosity of liquids decreases, while that of gases increases. Because the viscosity of a liquid is primarily caused by the cohesive forces between its molecules; as the temperature rises, these cohesive forces weaken, resulting in a decrease in viscosity ； Gas viscosity arises from the thermal motion of gas molecules; as the temperature increases, this thermal motion intensifies, resulting in an increase in viscosity. All real fluids possess viscosity; they are known as viscous fluids ； An ideal fluid is a fluid that has absolutely no viscosity (=0). Characteristics of the static pressure of a fluid: Feature 1: The stress in a stationary fluid has only a normal component (there is no relative motion between fluid particles, hence no shear stress), and it acts along the inward normal direction. Feature 2: In a stationary fluid, the magnitude of the static pressure at any point is independent of the direction in which it acts, and its value is the same everywhere. Isopresses have the following two important properties: Property 1: In a fluid at equilibrium, the isopressure passing through any given point must be perpendicular to the body force acting on that point. Feature 2: When two immiscible liquids are in equilibrium, their interface must be a level of constant pressure. Principle for determining isopressures: In a gravitational field, within stationary, homogeneous, and continuous fluid, horizontal surfaces are isopressures. Absolute pressure: with a perfect vacuum as the zero point, denoted as p ； Relative pressure: With the local atmospheric pressure pa as the zero point, it is denoted as pg. The relationship between the two is: p=pg+pa ； Vacuum degree: The absolute value of the relative pressure when it is negative is known as the vacuum pressure. Conclusion on the force exerted by a stationary liquid on a wall surface: 1. The average value of the hydrostatic pressure on a plane is the pressure at the centroid of the surface (the planar figure). The total pressure is equal to the pressure pC at the centroid C of the surface, multiplied by the area A of that surface. 2. For a force that is evenly distributed over a plane, the point of application of the resultant force is its centroid; however, the static pressure distribution is uneven – the greater the depth below the liquid surface, the higher the pressure. Therefore, the point of application of the total pressure lies below the centroid of the surface. 3. In the calculations, pressure is taken as relative pressure. Methods for studying fluid motion: the Lagrangian method and the Eulerian method. The Lagrange method focuses on fluid particles ； The Euler method focuses on spatial points in the flow field. Flow in which the flow parameters at various points in the flow field are independent of time is called steady-flow. Flow in which the flow parameters at various points in the flow field change over time is called unsteady flow. Streamlines are the lines of velocity direction at successive points in a flow field at the same instant. There are two characteristics of streamlines: 1) In unsteady flow, the shape of the streamlines changes over time ； In steady flow, there is no change over time; at this point, the streamlines coincide with the trajectories. 2) A streamline is a smooth curve, and streamlines cannot intersect; if they do intersect, the velocity ratio at the intersection point is zero. A trajectory is the path followed by a fluid particle as it moves. Streamlines have the following two characteristics: ① In unsteady flow, the shape of the streamlines changes over time ； In steady flow, its shape does not change over time. At this point, the streamlines coincide with the trajectories, and the fluid particles move along the streamlines. ②A streamline is a smooth curve. Streamlines cannot intersect. If they intersect, the velocity at the intersection point must be zero. Otherwise, two velocities would exist at the intersection point at the same time, which is clearly impossible. Wet perimeter: The length over the flow cross-section at which the fluid is in contact with the solid, denoted by χ. Hydraulic radius: The ratio of the total area A flowing through the cross-section to the wet perimeter χ, denoted by R. Hydraulic diameter: The value that is 4 times the hydraulic radius is called the hydraulic diameter. di=4A/χ=4R. System: A collection of numerous fluid particles is called a system. Once the system is determined, all the fluid particles it contains are also determined. The size, position, and shape of the system can be changed. Control volume: A control volume refers to a specific region in the flow field. The boundary of this space is called the control surface. Once the control body is selected, its size, position, and shape in a certain coordinate system remain unchanged. Physical meaning of the total flow continuity equation: The mass flow rate passing through any two cross-sections along the total flow path is equal, as expressed by this equation. Bernoulli’s equation has the following physical and geometric meanings: Physically, under certain specified conditions, the mechanical energy of a unit weight of fluid (potential energy, pressure energy, and kinetic energy) can be converted into one another, but their total remains constant. Geometric meaning—under the given constraints, the total head along the same streamline is a constant. The total mechanical energy remains constant, but it is not the case that the energy of each individual component stays constant. The three forms of energy can increase or decrease individually and transform into one another, but their total amount remains constant. Euler's view: In the steady flow of an ideal fluid, the unit total mechanical energy is equal for any two fluid particles located on the same streamline. Lagrange’s view: In the steady flow of an ideal fluid, the total mechanical energy per unit mass remains constant for the same fluid particle. Head line: Represents the changes in various heads along a path in a geometric manner. The total head line of a steady one-dimensional flow in an ideal fluid is horizontal. When the radius of curvature of the streamlines is large or the angle between the fluids is small, the streamlines are approximately parallel straight lines; such flow is called a gradually varying flow, otherwise it is called a rapidly varying flow. Characteristics of gradually varying flow: The distribution law of the hydrostatic pressure at any cross-section under gradually varying flow is the same as that in a steady-flow regime, with z + p/ρg = constant. Conditions for applying the momentum equation: ① Steady flow. ②The fluid is incompressible. Reasons for frictional loss: It arises from the friction between the fluid and the wall surface. Reasons for local loss: It results from the impact of the fluid against the wall surface as well as from collisions between fluid particles. Criterion for determining flow regime – Reynolds number. Hydraulically smooth pipes vs. hydraulically rough pipes: In turbulent flow, there is a laminar sublayer; when the thickness of this laminar sublayer δl is greater than 5Δ, the roughness of the pipe surface is almost completely covered by this laminar sublayer, and the wall surface has little influence on the fluid in the turbulent region. This situation is similar to fluid flow in a perfectly smooth pipe, and such pipes are referred to as hydraulically smooth pipes. When the thickness of the laminar sublayer δl is less than 0.3Δ, almost all the protrusions on the pipe wall are exposed to turbulence; the fluid particles in the turbulent flow collide with these protrusions, resulting in an increase in resistance. In this case, the pipe is referred to as a hydraulically rough pipe. Nikolaus experiment: Zone I – laminar flow region, dependent only on the Reynolds number. Zone II — the first transition zone, with no obvious pattern. Zone III – hydrodynamically smooth zone, depending only on the Reynolds number. Zone IV — the second transition zone, related to relative roughness and Reynolds number. Zone V – hydraulic roughness zone, independent of the Reynolds number. Converting the loss of a local device to the frictional loss of a straight pipe of length le, this length le is the equivalent pipe length of that local device. Long pipes: Pipes in which the proportion of local losses to total losses is relatively small, such as <5%; in such cases, local losses are often ignored. Short pipe: A pipeline in which the frictional loss and local losses are of similar magnitude, and both need to be taken into account. For simple piping equations, the characteristic of series piping is that the flow rate in each pipe is equal, and this value is equal to the total flow rate ； The sum of the head losses in each pipe equals the total loss of the pipeline. The characteristic of parallel pipelines is that the sum of the flow rates in each pipeline equals the total flow rate ； The head losses in each pipe are equal, which equals the total loss of the pipeline. Similar conditions: geometric similarity, kinematic similarity, and dynamic similarity. Under the aforementioned similar conditions, geometric similarity is a necessary prerequisite, dynamic similarity is a decisive condition, and kinematic similarity is an inevitable result of geometric and dynamic similarity. The property of a physical quantity’s unit is called dimension; dimensions are further divided into fundamental dimensions and derived dimensions ； Fundamental dimensions are independent; for example, in dynamic problems that are independent of temperature, length, time, and mass can be chosen as the fundamental dimensions. Dimensional consistency of physical equations — In any correct physical equation, the dimensions of all terms must be the same. Characteristic parameters of the pump: (1) Flow rate Q – The volume of liquid that passes through the pump per unit of time is called the pump’s flow rate, also known as displacement, and is measured in m3/min. (2) Head H — The total energy gained per unit weight of liquid inside the pump is called the pump’s head, measured in meters. (3) Rotational speed n — The number of revolutions per minute of the pump impeller is called the rotational speed, with the unit being r/min. (4) Power — Pump power is divided into shaft power and useful power. ①Shaft power N is the power transmitted by the prime mover to the pump shaft, measured in W or kW. ②Effective power Na: The actual energy obtained by the liquid from the pump per unit of time is known as the pump’s effective power, which is measured in watts or kilowatts. (5) Efficiency η—the ratio of the pump’s effective power to its shaft power is called efficiency. (6) Allowable suction vacuum level—this parameter indicates the pump’s ability to draw in liquid, measured in m. Characteristic parameters of the fan: (1) Flow rate Q – The volume of gas that passes through the fan per unit of time is called the fan’s flow rate, also known as air volume; its unit is m3/min. (2) Pressure P — Pressure includes total pressure and static pressure. The total energy obtained per unit volume of gas within the fan is called the fan total pressure P, with the unit being Pa ； The static pressure Pst of a fan, which is the total pressure of the fan minus the dynamic pressure at the fan outlet, is measured in Pa. (3) Rotational speed n — The number of revolutions per minute of the fan impeller is called the rotational speed, with the unit being r/min. (4) Power — The power of a ventilator includes shaft power and useful power. ①Shaft power N — the power transmitted by the prime mover to the fan shaft, measured in W or KW. ②Effective power Na – the actual energy obtained by the gas from the fan per unit of time, measured in W or kW. (5) Efficiency—The ratio of the fan’s useful power to its shaft power is called the fan’s efficiency. Theoretical flow rate: The flow rate without considering leaks. Conclusion of the basic equation for the theoretical head when there are an infinite number of blades: ① The energy obtained per unit weight of fluid depends only on the velocities of the fluid at the inlet and outlet of the blades, and is independent of the flow process. ②The energy obtained per unit weight of fluid is independent of the type of fluid being transported. In other words, whether the fluid being transported is a liquid or a gas, as long as the velocity triangles at the inlet and outlet of the blade are the same, the same head can be obtained. ③The energy obtained per unit weight of fluid is proportional to the circumferential velocity u2 at the outer edge of the impeller, and u2=πnD2/60. Therefore, when all other conditions are equal, the larger the outer diameter D2 of the impeller, the higher the rotational speed n, and the greater the head pressure. With the same size and speed, impellers with forward-curving blades achieve the highest theoretical head, those with radial blades come next, while impellers with backward-curving blades yield the lowest head. There are various losses when centrifugal pumps or fans are in operation. Based on their causes, they can be divided into three types: hydraulic loss, volumetric loss, and mechanical loss. Cavitation: Material damage caused by the formation, growth, and collapse of bubbles within a fluid flow due to pressure changes is known as cavitation. Cavitation hazards: (1) Material damage (2) Noise and vibration (3) Decline in performance. Normal and appropriate operating conditions for pumps: 1. Stable operating conditions; 2. The operating point lies within the industrially acceptable range; 3. The net positive suction head available in the actual installation is greater than the pump’s allowable net positive suction head. Adjustment of the pump’s operating points: 1. Throttling regulation; 2. Reducing the number of impellers; 3. Decreasing the diameter of the impellers. Starting the pump: 1. Inspect the pump to ensure that all components are firmly connected, that the pump shaft rotates smoothly, and that the suction filter is not clogged; then rotate the pump to verify that it moves freely without any sticking. 2. Fill the pump chamber and suction pipe with water, remove the air from the pump chamber, close the stop valve on the drain pipe, and then start the motor. 3. When the pump reaches its normal operating speed and the reading on the ammeter returns to normal, gradually open the shut-off valve and set it at an appropriate opening degree to enable normal operation. Function of the diffuser: To recover some of the dynamic pressure in order to increase the static pressure of the fan unit. Surge: When a fan with a cambered or saddle-shaped pressure curve operates in a network containing large containers, it can cause the fan’s flow rate to experience sudden and drastic changes, leading to intense mechanical vibrations; this phenomenon is known as surge. The selection of the starting operating condition should be considered from two aspects: the power at the starting operating point should be minimal ； Avoid instability during startup. Therefore, for centrifugal fans whose wind pressure characteristic curve decreases monotonically, wall dampers should be activated to minimize the starting power. For fans with characteristic curves that are hump-shaped or saddle-shaped, in order to prevent the operating point from passing through the unstable region during startup, the dampers should be opened partially or fully at startup. There are two methods for adjusting fans: one is to change the characteristics of the system, such as through damper adjustment, and the other is to adjust the characteristics of the fan itself, such as by changing its speed, using a diffuser, altering the number of impeller stages, or changing the number of blades.