{"id":14207,"date":"2017-04-17T17:37:22","date_gmt":"2017-04-17T17:37:22","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=14207"},"modified":"2022-10-22T08:08:02","modified_gmt":"2022-10-22T08:08:02","slug":"continuity-equation","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-engineering\/fluid-dynamics\/continuity-equation\/","title":{"rendered":"Continuity Equation"},"content":{"rendered":"
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The continuity equation<\/strong> is simply a mathematical expression of the principle of conservation of mass. For a control volume with a single inlet<\/strong> and a single outlet<\/strong>, the principle of conservation of mass states that, for steady-state flow<\/strong>, the mass flow rate into the volume must equal the mass flow rate out.<\/p>\n

m\u0307in<\/sub> = m\u0307out<\/sub>\u00a0<\/span><\/p>\n

Mass entering per unit time = Mass leaving per unit time<\/p>\n<\/div><\/div>\n

Conservation of Mass<\/h2>\n
The mass can neither be created nor destroyed.<\/div><\/div>\n

This principle is generally known as the conservation of matter principle<\/a><\/strong>. It states that the mass of an object or collection of objects never changes over time, no matter how the constituent parts rearrange themselves. This principle can be used in the analysis of flowing fluids<\/strong>. Conservation of mass in fluid dynamics<\/strong><\/a> states that all mass flow rates into<\/strong> a control volume are equal to all mass flow rates out<\/strong> of the control volume plus the rate of mass change within the control volume.<\/p>\n

\"Continuity<\/a>
Continuity Equation – Definition<\/figcaption><\/figure>\n

This principle is expressed mathematically by the following equation:<\/p>\n

m\u0307in<\/sub> = m\u0307out<\/sub> +\u2206m<\/sup>\u2044\u2206t<\/sub><\/span><\/p>\n

Mass entering per unit time = Mass leaving per unit time + Increase of mass in the\u00a0control volume per unit time<\/p>\n

This equation describes nonsteady-state flow<\/strong>. Nonsteady-state flow refers to the condition where the fluid properties at any single point in the system may change over time. Steady-state flow<\/strong> refers to the condition where the fluid properties (temperature, pressure, and velocity<\/strong>) at any single point in the system do not change over time<\/strong>. But one of the most significant constant properties in a steady-state flow system is the system mass flow rate. This means that there is no accumulation<\/strong> of mass within any component in the system.<\/p>\n

Continuity Equation<\/h2>\n
\"Continuity<\/a>
Example of flow rates in a reactor. It is an illustrative example, and the data do not represent any reactor design.<\/figcaption><\/figure>\n

The continuity equation<\/strong> is simply a mathematical expression of the principle of conservation of mass. For a control volume with a single inlet<\/strong> and a single outlet<\/strong>, the principle of conservation of mass states that, for steady-state flow<\/strong>, the mass flow rate into the volume must equal the mass flow rate out.<\/p>\n

m\u0307in<\/sub> = m\u0307out<\/sub>\u00a0<\/span><\/p>\n

Mass entering per unit time = Mass leaving per unit time<\/p>\n

This equation is called the continuity equation<\/strong> for steady one-dimensional flow. The net mass flow must be zero for a steady flow through a control volume with many inlets and outlets, where negative inflows and outflows are positive.<\/p>\n

This principle can be applied to a stream tube<\/b>\u00a0such as that shown above. No fluid flows across the boundary made by the streamlines,<\/strong> so mass only enters and leaves through the two ends of this stream tube section.<\/p>\n

When a fluid is in motion, it must move in such a way that mass is conserved. To see how mass conservation places restrictions on the velocity field, consider the steady flow of fluid through a duct (that is, the inlet and outlet flows do not vary with time).<\/p>\n

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Differential Form of Continuity Equation<\/h2>\n

A general continuity equation can also be written in a differential form<\/strong>:<\/p>\n

\u2202\u2374<\/sup>\u2044\u2202t<\/sub> + \u2207 . (\u2374 \u035ev) = \u03c3<\/span><\/p>\n

where<\/strong><\/p>\n