Transformation formula

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The law of the conservation of mass dictates that mass may be neither created nor destroyed. For a control volume that contains only one phase, conservation of mass can be obtained by setting the general and specific property forms to <math>\Phi  = m</math> and <math>\phi  = 1</math> in eq. (2.3), i.e.,
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[[Image:Gov_(1).jpg|thumb|400 px|alt=Relation between a fixed mass system and a control volume.|<center>'''Relation between a fixed mass system and a control volume.'''</center>]]
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<center><math>{\left. {\frac{{dm}}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} }     \qquad \qquad(1)</math></center>
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A fixed-mass system describes an amount of matter that can move, flow and interact with the surroundings, but the control volume approach depicts a region or volume of interest in a flow field, which is not unique and depends on the user. So conservation laws for a fixed-mass system need to be transformed to apply to a control volume. A mathematical relation that allows one to mathematically and physically link the conservation laws for a control volume with that of a fixed-mass system will be derived.
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The figure on the right shows the flow field under consideration. At time <big><math>t</math></big>, the control volume shown by the solid line coincides with a single-phase fixed-mass system depicted by the dashed line. At time <big><math>t+dt</math></big>, a portion of the fixed-mass system moves outside of the boundaries of the control volume. It can be seen from the figure on the right that region I is occupied by the system at time <big><math>t</math></big> only, region II is common to the system at both <big><math>t</math></big> and <big><math>t+dt</math></big>, and region III is occupied by the system at <big><math>t+dt</math></big> only. For a system with a fixed-mass, the change of the general property <big><math>\Phi</math></big>, which has a specific value (per unit mass) <big><math>\phi</math></big>, can be written as [[#References|(Welty, 1978)]]:
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<center><math>{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{{\Phi {{\left| {_{t + dt} - \Phi } \right|}_t}}}{{dt}}    \qquad \qquad(1)</math></center>
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Considering that the control mass occupies regions I and II at time <math>t</math>, one obtains <math>{\left. \Phi  \right|_t} = {\left. \Phi  \right|_I} + {\left. \Phi  \right|_{II}}.</math> At time <math>t+dt</math>, the fixed-mass system occupies both regions II and III, i.e., <math>{\left. \Phi  \right|_{t + dt}} = {\left. \Phi  \right|_{II}} + {\left. \Phi  \right|_{III}}.</math> Therefore, eq. (1) can be rewritten as
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<center><math>{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{{{\Phi _{II}}\left| {_{t + dt} - {\Phi _{II}}\left| {_t} \right.} \right.}}{{dt}} + \frac{{{\Phi _{III}}\left| {_{t + \Delta t}} \right.}}{{dt}} - \frac{{{\Phi _I}\left| {_t} \right.}}{{dt}}    \qquad \qquad(2) </math></center>
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where the first term on the right hand side represents the time rate of change of the mass of the contents of the control volume, and the second term on the right hand side represents the net rate of mass flow through the control surface. The term <math>({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA</math> in the mass flow integral represents the product of the velocity component perpendicular to the control surface and differential area. This term is the volume flowrate through <math>dA</math>, and becomes the mass flowrate when multiplied by density, ''ρ''.
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The first term on the right-hand side of eq. (2) may be written as 
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<center><math>\frac{{{\Phi _{II}}\left| {_{t + dt} - {\Phi _{II}}\left| {_t} \right.} \right.}}{{dt}} = \frac{{d\Phi {|_{II}}}}{{dt}}</math></center>
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and represents the rate of change of property <big><math>\Phi</math></big> within the control volume, because region II becomes coincident with the control volume as <big><math>dt \to 0</math></big>. For the general case of variable <big><math>\Phi</math></big> within a control volume, it is appropriate to write the time derivative of <big><math>\Phi</math></big> as
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Since the mass of a fixed-mass system is constant by definition, and the fixed-mass system contains only one phase, the resulting formulation of conservation of mass is  
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<center><math>\frac{{d\Phi {|_{II}}}}{{dt}} = {\left( {\frac{{d\Phi }}{{dt}}} \right)_{CV}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV} </math></center>
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<center><math>\frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} } = 0     \qquad \qquad(2)</math></center>
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where <math>V</math> is the volume of the control volume.
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[[Image:Gov_(2).jpg|thumb|400 px|alt=Control volume in a flow field|<center>'''Control volume in a flow field.'''</center>]]
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The second and third terms on the right-hand side of eq. (2), respectively, represent the property <big><math>\Phi</math></big> leaving and entering the control volume due to mass flow across its boundary. If the absolute velocity is <math>V</math>, and the reference frame moves with a constant velocity <big><math>{V_{ref}}</math></big>, the relative velocity is <math>{{\mathbf{V}}_{rel}} = {\mathbf{V}} - {{\mathbf{V}}_{ref}}</math>. For the control volume’s entire surface area, <big><math>A</math></big>, the rate of movement of property <big><math>\Phi</math></big> due to mass flow may be written as (see figure on the right)
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<center><math>\frac{{{\Phi _{III}}\left| {_{t + \Delta t}} \right.}}{{dt}} - \frac{{{\Phi _I}\left| {_t} \right.}}{{dt}} = \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA} </math></center>
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where <big><math>n</math></big> is the normal direction of the control volume.
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Therefore, eq. (2) can be rewritten as
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<center><math>{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA} }    \qquad \qquad(3)</math></center>
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which is the final form of the transformation formula that relates the change of property for a fixed-mass system to that of the control volume. It states that the rate of change of a property <big><math>\Phi</math></big> for a fixed-mass system is equal to the rate of change of <big><math>\Phi</math></big> in the control volume (the first term on the right-hand side) plus the net rate of efflux of <big><math>\Phi</math></big> by mass flow into or out of the control volume (the second term on the right-hand side).
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which shows that the time rate of change of the mass of the contents of the control volume plus the net rate of mass flow through the control surface must equal zero. In other words, the sum of the mass flow rate into and out of the control volume must be equal to the accumulation and depletion of the mass within the control volume.
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It should be pointed out that the control volume moves with the reference frame, which moves with a constant velocity, <big><math>{V_{ref}}</math></big>. The coordinate system is attached to and moves with the reference frame. In other words, the coordinate system is stationary relative to the reference frame. The reference frame is inertial, so Newton’s second law is valid in the coordinate system that moves with the reference frame. Equation (3) will be used to obtain the macroscopic (integral) formulation of the basic laws for a control volume.
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For a control volume containing multiple phases separated by interfaces, the conservation of mass can be similarly obtained (Faghri and Zhang, 2006):
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==References==
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<center><math>\sum\limits_{k = 1}^\Pi  {\left[ {\frac{\partial }{{\partial t}}\int_{{V_k}(t)} {{\rho _k}dV}  + \int_{{A_k}(t)} {{\rho _k}({{\mathbf{V}}_{k,rel}} \cdot {{\mathbf{n}}_k})dA} } \right]}  = 0    \qquad \qquad(3)</math></center>
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Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Elsevier, Burlington, MA
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where <math>k</math> denotes the <math>k</math><sup>th</sup> phase in the multiphase system, and <math>\Pi </math> is the number of phases.  
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Faghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced  Heat and Mass Transfer'', Global Digital Press, Columbia, MO.
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==References==
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Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Burlington, MA.
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Welty, J.R., 1978, ''Engineering Heat Transfer'', John Wiley & Sons, New York.  
==Further Reading==
==Further Reading==
==External Links==
==External Links==

Current revision as of 13:04, 14 July 2010

Relation between a fixed mass system and a control volume.
Relation between a fixed mass system and a control volume.

A fixed-mass system describes an amount of matter that can move, flow and interact with the surroundings, but the control volume approach depicts a region or volume of interest in a flow field, which is not unique and depends on the user. So conservation laws for a fixed-mass system need to be transformed to apply to a control volume. A mathematical relation that allows one to mathematically and physically link the conservation laws for a control volume with that of a fixed-mass system will be derived.

The figure on the right shows the flow field under consideration. At time t, the control volume shown by the solid line coincides with a single-phase fixed-mass system depicted by the dashed line. At time t + dt, a portion of the fixed-mass system moves outside of the boundaries of the control volume. It can be seen from the figure on the right that region I is occupied by the system at time t only, region II is common to the system at both t and t + dt, and region III is occupied by the system at t + dt only. For a system with a fixed-mass, the change of the general property Φ, which has a specific value (per unit mass) φ, can be written as (Welty, 1978):

{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{{\Phi {{\left| {_{t + dt} - \Phi } \right|}_t}}}{{dt}}     \qquad \qquad(1)

Considering that the control mass occupies regions I and II at time t, one obtains {\left. \Phi  \right|_t} = {\left. \Phi  \right|_I} + {\left. \Phi  \right|_{II}}. At time t + dt, the fixed-mass system occupies both regions II and III, i.e., {\left. \Phi  \right|_{t + dt}} = {\left. \Phi  \right|_{II}} + {\left. \Phi  \right|_{III}}. Therefore, eq. (1) can be rewritten as

{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{{{\Phi _{II}}\left| {_{t + dt} - {\Phi _{II}}\left| {_t} \right.} \right.}}{{dt}} + \frac{{{\Phi _{III}}\left| {_{t + \Delta t}} \right.}}{{dt}} - \frac{{{\Phi _I}\left| {_t} \right.}}{{dt}}     \qquad \qquad(2)

The first term on the right-hand side of eq. (2) may be written as

\frac{{{\Phi _{II}}\left| {_{t + dt} - {\Phi _{II}}\left| {_t} \right.} \right.}}{{dt}} = \frac{{d\Phi {|_{II}}}}{{dt}}

and represents the rate of change of property Φ within the control volume, because region II becomes coincident with the control volume as dt \to 0. For the general case of variable Φ within a control volume, it is appropriate to write the time derivative of Φ as

\frac{{d\Phi {|_{II}}}}{{dt}} = {\left( {\frac{{d\Phi }}{{dt}}} \right)_{CV}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV}

where V is the volume of the control volume.

Control volume in a flow field
Control volume in a flow field.

The second and third terms on the right-hand side of eq. (2), respectively, represent the property Φ leaving and entering the control volume due to mass flow across its boundary. If the absolute velocity is V, and the reference frame moves with a constant velocity Vref, the relative velocity is {{\mathbf{V}}_{rel}} = {\mathbf{V}} - {{\mathbf{V}}_{ref}}. For the control volume’s entire surface area, A, the rate of movement of property Φ due to mass flow may be written as (see figure on the right)

\frac{{{\Phi _{III}}\left| {_{t + \Delta t}} \right.}}{{dt}} - \frac{{{\Phi _I}\left| {_t} \right.}}{{dt}} = \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA}

where n is the normal direction of the control volume. Therefore, eq. (2) can be rewritten as

{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA} }     \qquad \qquad(3)

which is the final form of the transformation formula that relates the change of property for a fixed-mass system to that of the control volume. It states that the rate of change of a property Φ for a fixed-mass system is equal to the rate of change of Φ in the control volume (the first term on the right-hand side) plus the net rate of efflux of Φ by mass flow into or out of the control volume (the second term on the right-hand side).

It should be pointed out that the control volume moves with the reference frame, which moves with a constant velocity, Vref. The coordinate system is attached to and moves with the reference frame. In other words, the coordinate system is stationary relative to the reference frame. The reference frame is inertial, so Newton’s second law is valid in the coordinate system that moves with the reference frame. Equation (3) will be used to obtain the macroscopic (integral) formulation of the basic laws for a control volume.

References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Welty, J.R., 1978, Engineering Heat Transfer, John Wiley & Sons, New York.

Further Reading

External Links