{"id":14378,"date":"2017-05-12T19:24:53","date_gmt":"2017-05-12T19:24:53","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=14378"},"modified":"2022-10-23T15:14:20","modified_gmt":"2022-10-23T15:14:20","slug":"momentum-formula-momentum-equation","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-engineering\/fluid-dynamics\/conservation-momentum-fluid-dynamics\/momentum-formula-momentum-equation\/","title":{"rendered":"Momentum Formula – Momentum Equation"},"content":{"rendered":"
<\/div>\n

We assume fluid to be both steady<\/strong> and incompressible<\/strong>. To determine the rate of change of momentum for a fluid, we will consider a stream tube (control volume<\/strong>) as we did for the Bernoulli equation<\/a>. In this control volume,<\/strong> any change in momentum of the fluid within a control volume is due to the action of external forces on the fluid within the volume.<\/p>\n

\"Conservation<\/a>As can be seen from the picture, the control volume method<\/strong> can analyze the law of conservation of momentum in a fluid. A control volume is an imaginary surface<\/strong> enclosing a volume of interest. The control volume can be fixed or moving, and it can be rigid or deformable. To determine all forces acting on the surfaces of the control volume, we have to solve the conservation laws in this control volume.<\/p>\n

The first conservation equation we have to consider in the control volume is the continuity equation<\/a> (the law of conservation of matter<\/a>). In the simplest form, it is represented by the following equation:<\/p>\n

\u2211m\u0307in<\/sub> = \u2211m\u0307out<\/sub> <\/span><\/p>\n

Sum of mass flow rates entering per unit time = Sum of mass flow rates leaving per unit time<\/p>\n

The second conservation equation we have to consider in the control volume is the momentum formula<\/strong>. <\/span><\/strong>In the simplest form, the momentum formula<\/strong> can be represented by the following equation:<\/p>\n

–
\n\"Momentum<\/a><\/p>\n

<\/h2>\n

Choosing a Control Volume<\/h2>\n

A control volume can be selected as any arbitrary volume through which fluid flows. This volume can be static, moving, and even deforming during flow. To solve any problem, we have to solve basic conservation laws in this volume. It is very important to know all relative flow velocities to the control surface. Therefore it is very important to define exactly the boundaries of the control volume during an analysis.<\/span><\/strong>

<\/div><\/p><\/div><\/div>
<\/div>\n

Example: The force acting on a deflector elbow<\/h2>\n

\"Momentum<\/a>An elbow<\/strong> (let say of primary piping) is used to deflect water flow at a velocity of 17 m\/s<\/strong>. The piping diameter is equal to 700 mm<\/strong>. The gauge pressure inside the pipe is about 16 MPa<\/strong> at a temperature of 290\u00b0C. Fluid is of constant density \u2374 ~ 720 kg\/m3<\/sup><\/strong> (at 290\u00b0C). The angle of the elbow is 45\u00b0<\/strong>.<\/p>\n

Calculate the force on the wall<\/strong> of a deflector elbow (i.e., calculate vector F3<\/sub>).<\/p>\n

Assumptions:<\/strong><\/p>\n