Transport phenomena (Continuity Equation)
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This document discusses the continuity equation in three dimensions. It begins by stating that the continuity equation is based on the principle of conservation of mass. It then presents the mathematical form of the continuity equation in three dimensions, showing how it accounts for the rate of increase of mass in an infinitesimal fluid element equaling the net rate of mass flow into and out of the element. The document provides explanations and derivations of the terms in the three-dimensional continuity equation. It concludes by noting some simplified forms of the equation under certain conditions like steady state or incompressible flow.
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2. Streamlines, pathlines, and streaklines are defined. Streamlines give the instantaneous flow pattern. Continuity equations are derived for compressible, incompressible, 1D, and 2D steady flows.
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Transport phenomena (Continuity Equation)
- 1. Transport Phenomena CONTINUITY EQUATION IN 3 DIMENSIONS:
- 2. 2 HELLO! I’m Mujeeb UR Rahman Chemical Engineering student @Mehran University of Engineering & Technology Jamshoro, Pakistan. You can find me at SlideShare @MujeebURRahman38 ResearchGate @Mujeeb UR Rahman, Academia @Mujeeb UR Rahman
- 3. Transport Phenomena 3 CONTINUITY EQUATION IN 3 DIMENSIONS: “The equation based on the principle of conversation of mass.” A E C D G F H B dx dz 2kg 1kg 1kg x-axis (u) y-axis (v) z-axis (w)
- 4. 4 ▪ Law of conservation of mass CONTINUITY EQUATION IN 3 DIMENSIONS: 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 = 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑖𝑛 − 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑚𝑎𝑠𝑠 𝑜𝑢𝑡 Transport Phenomena
- 5. Transport Phenomena 5 CONTINUITY EQUATION IN 3 DIMENSIONS: Direction of x, y, z. u inlet velocity components in x direction. v inlet velocity components in y direction. w inlet velocity components in z direction. Mass of fluid entering the face +ve face ABCD (inflow) per second . 𝑚 𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝜌 = 𝑚 𝑉 𝑚 = 𝜌 × 𝑉 𝑚 𝑠 𝜌 × 𝑉 𝑠 𝜌 × 𝐴 × 𝑙 𝑠 𝜌 × 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐴𝐵𝐶𝐷 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑛 𝑥 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 = 𝜌𝐴𝑉 𝜌 × 𝑢 × (𝑑𝑦 × 𝑑𝑧)
- 6. 6 Mass of fluid leaving the face EFGH (outflow) per second, CONTINUITY EQUATION IN 3 DIMENSIONS: 𝜌𝑢 𝑑𝑦𝑑𝑧 + 𝜕 𝜕𝑥 𝜌𝑢 𝑑𝑦𝑑𝑧 𝑑𝑥 Rate of increase in mass x-direction = mass through ABCD – mass through EFGH = 𝜌𝑢 𝑑𝑦𝑑𝑧 − 𝜌𝑢 𝑑𝑦𝑑𝑧 − 𝜕 𝜕𝑥 𝜌𝑢 𝑑𝑦𝑑𝑧 𝑑𝑥 = − 𝜕 𝜕𝑥 𝜌𝑢 𝑑𝑦𝑑𝑧𝑑𝑥 = − 𝜕 𝜕𝑥 𝜌𝑢 𝑑𝑥𝑑𝑦𝑑z Transport Phenomena
- 7. 7 Similarly, Rate of increase in mass in y-direction = − 𝜕 𝜕𝑦 𝜌𝑣 𝑑𝑥𝑑𝑦𝑑z Rate of increase in mass in y-direction = − 𝜕 𝜕𝑧 𝜌𝑤 𝑑𝑥𝑑𝑦𝑑z ➢ Total rate of increase in mass = − 𝜕 𝜕𝑥 𝜌𝑢 + 𝜕 𝜕𝑦 𝜌𝑣 + 𝜕 𝜕𝑧 𝜌𝑤 𝑑𝑥𝑑𝑦𝑑𝑧 CONTINUITY EQUATION IN 3 DIMENSIONS: By the law of conversation of mass, there is no accumulation of mass i.e mass is neither be created nor destroyed in the fluid element. So net increase of mass per unit time in the fluid element must be equal to the rate of increase of mass of fluid in the element. ...(1) Transport Phenomena
- 8. 8 CONTINUITY EQUATION IN 3 DIMENSIONS: Mass of fluid in the element is = 𝜌 𝑑𝑥𝑑𝑦𝑑𝑧 𝜌 = 𝑚 𝑉 𝑚 = 𝜌𝑉 its rate of increase with time = 𝜕 𝜕𝑡 𝜌 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝜕𝜌 𝜕𝑡 𝑑𝑥𝑑𝑦𝑑𝑧 Equating eq. (i) and (ii) − 𝜕 𝜕𝑥 𝜌𝑢 + 𝜕 𝜕𝑦 𝜌𝑣 + 𝜕 𝜕𝑧 𝜌𝑤 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝜕𝜌 𝜕𝑡 𝑑𝑥𝑑𝑦𝑑𝑧 …(2) Transport Phenomena
- 9. 9 CONTINUITY EQUATION IN 3 DIMENSIONS: 𝜕𝜌 𝜕𝑡 = − 𝜕 𝜕𝑥 𝜌𝑢 + 𝜕 𝜕𝑦 𝜌𝑣 + 𝜕 𝜕𝑧 𝜌𝑤 This is the equation of continuity, which describes the time rate of change of the fluid density at a fixed point in space. This equation can be written more concisely by using vector notation as follows ) . ( V t − = Rate of increase of mass per unit volume Net rate of addition of mass per unit volume by convection Transport Phenomena
- 10. 𝜕𝜌 𝜕𝑡 + 𝜕 𝜕𝑥 𝜌𝑢 + 𝜕 𝜕𝑦 𝜌𝑣 + 𝜕 𝜕𝑧 𝜌𝑤 = 0 ▪ For steady state flow 𝜕𝜌 𝜕𝑡 = 0 and hence above equation becomes as, 𝜕 𝜕𝑥 𝜌𝑢 + 𝜕 𝜕𝑦 𝜌𝑣 + 𝜕 𝜕𝑧 𝜌𝑤 = 0 ▪ If the fluid is incompressible, then 𝜌 is constant and the above eq. becomes as, 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + 𝜕𝑤 𝜕𝑧 = 0 CONTINUITY EQUATION IN 3 DIMENSIONS: Continuity equation in three-dimensions. Transport Phenomena
- 11. Thanks! 11