13.17
連続性の方程式
流体の動きは、速度ベクトルまたは流線のいずれかで表されます。 一定期間内に特定の場所を通過して領域を通過する流体の体積は、 流量Q、またはより正確には体積流量と呼ばれます。 流量と速度は関連しています。たとえば、川の水の速度が大きいほど、川の流量は大きくなります。 ただし、流量は川の大きさや形によっ…
単位時間内にエリア内のポイントを流れる流体の体積が、体積流量を示します。
体積を面積×距離に置き換え、速度と距離の関係を使用すると、体積流量は面積と流体の速度の積に等しくなります。
不規則な断面パイプを着実に流れるすべての点で同じ密度の非圧縮性流体を考えてみましょう。定常的な流れの場合、ある点での流体の速度と密度は時間とともに一定に保たれます。
単位時間あたりにポイントを通過する流体の質量は質量流量と呼ばれ、密度に体積流量を掛けたものに等しくなります。
パイプには他のソースまたはシンクがないため、パイプに流れ込む質量は、パイプから出る質量と等しくなければなりません。これにより、流体の連続性の一般的な方程式が得られます。
非圧縮性流体の場合、密度は相殺されます。したがって、パイプへの体積流量は、パイプからの体積流量と等しくなります。
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Q1: What is volume flow rate and how is it calculated?
Volume flow rate is the volume of fluid flowing through a point in an area per unit time. It is calculated by multiplying the cross-sectional area of the pipe by the velocity of the fluid. This relationship shows that flow rate depends on both the size of the conduit and how fast the fluid moves through it.
Q2: Why does fluid velocity increase when a pipe narrows?
When a pipe's cross-sectional area decreases, the fluid velocity must increase to maintain continuity of flow. Since the same amount of incompressible fluid must pass through any point in the pipe per unit time, a smaller area requires higher velocity to conserve the volume flow rate.
Q3: What is the equation of continuity for incompressible fluids?
The equation of continuity states that the volume flow rate entering a pipe equals the volume flow rate leaving it. For incompressible fluids, density cancels out, so the product of cross-sectional area and velocity remains constant throughout the pipe, ensuring mass conservation.
Q4: How does mass flow rate relate to volume flow rate?
Mass flow rate equals density multiplied by volume flow rate. It represents the mass of fluid passing through a point per unit time. For steady flow through a pipe with no sources or sinks, the mass flowing in must equal the mass flowing out.
Q5: What conditions must be met for steady flow in a pipe?
For steady flow, the velocity and density of the fluid at any point must remain constant over time. Additionally, the pipe must have no sources or sinks that add or remove fluid, ensuring that mass entering the pipe equals mass leaving it at all times.
Q6: Why is the equation of continuity valid for liquids but not always for gases?
Liquids are essentially incompressible, so their density remains constant, making the equation of continuity universally valid for all liquid flow. Gases are compressible, so the equation must be applied with caution when gases undergo compression or expansion, as density changes affect flow relationships significantly.
Q7: How does the equation of continuity relate to energy conservation in fluid flow?
The equation of continuity ensures mass conservation in steady flow by maintaining constant volume flow rate through varying pipe sections. This principle connects to energy conservation and bernoulli equation principles, where changes in velocity and pressure are governed by both mass continuity and energy conservation in fluid systems.