1.2 Eulerian and Lagrangian Coordinates Eulerian coordinate
Source: Mass Transfer Fundamentals
1.2
Author: Dr. Yoram Cohen
Eulerian and Lagrangian Coordinates Eulerian coordinate - the independent variables are x, y, z and t or xi, t (i=1...3). This is a
fixed coordinate system. The basic conservation equations are in the Eulerian frame, R = R (xi,t). In the Lagrangian frame attention is fixed on a particular mass of fluid as it flows, R = R (xoi,t), where the coordinate xio specifies which fluid element is being considered.
1.3
Material Derivative Consider a variable " such that (1.2)
then the total differential of " can be expressed as
(1.3) division by a time differential *t leads to the following expression (1.4) After taking the limit *t 6 0 is obtained for the material derivative
(1.5)
in which vi is the fluid velocity in direction i. The material derivative (Lagrangian time derivative) represents the total change in " as seen by an observer who is moving with a particular fluid element In the Lagrangian frame we observe the particle for a time *t as it flows. The position of the particle changes by *xi while " changes by *".
-1-
Source: Mass Transfer Fundamentals
1.4
Author: Dr. Yoram Cohen
Control Volume Consider an arbitrary control volume,
(1.6) in which " is a fluid property, V is an arbitrary volume and L is a differential operator operating on
". The above integral equation can be satisfied for any arbitrary control volume only if L" = 0. Thus, as we will see later, Eq. (1.6) can be used to convert an integro-differential equation to a differential equation.
1.5
Reynolds Transport Theorem We begin by considering an arbitrary volume element, V, as it moves and deforms throug
a fluid domain. Fluid enters the volume element only through the open faces shown in the Figure
Figure 1.5.1 Arbitrary Fluid Volume Element As the volume element moves, volume-specific properties (e.g., density, enthalpy/volume) can change spatially and temporally. In order to describe the change of any specific volume-specific property, we can express the total time derivative of the selected property (integrated over the
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Source: Mass Transfer Fundamentals
Author: Dr. Yoram Cohen
volume element) using a finite difference approximation over an infinitesimal time interval, *t. Accordingly, the change of a volume specific property, ", with time can be expressed as
(1.7)
Equation 1.7 can be written in a more compact form
(1.8) The volume can change as the control volume moves in space during a specified time period. Thus, the differential control volume is determined by the fluid inflow into the element (i.e., the velocity perpendicular to the face of the element) velocity perpendicular to the area of the element (1.9) where ds is the surface area, v is the velocity and n is the unit normal. Using Eq. 1.9, the balance equation 1.8 can be written in the following form (1.10a)
(1.10b) In order to reduce the above equation to a more convenient form we make use of the "divergence theorem" (or Gauss' theorem): (1.11) -3-
Source: Mass Transfer Fundamentals
Author: Dr. Yoram Cohen
in which a is any vector and n is the unit normal. Therefore, the first integral term on the R.H.S of Eq. 1.10b can be written as
(1.12) and Eq. 1.10b becomes (1.13)
Another convenient form for Eq. 1.13 is in terms of index notations,
(1.14) Equations 1.13 and 1.14 are two alternative forms of the Reynolds' Transport Theorem. In Eqs. 1.13 and 1.14 the LHS represents the Lagrangian Derivative and the RHS represents the Eulerian Derivatives.
1.6
Conservation of Mass The equation for conservation of mass can be derive by considering the density of the
medium as the property of interest. The density is substituted for " in the Reynolds transport equation ( eq. 1.13) resulting in the following relation
(1.15)
Conservation of mass requires (excluding nuclear reactions) that the mass of the system remains unchanged, assuming that there are no nuclear reactions in the domain of interest.
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Source: Mass Transfer Fundamentals
Author: Dr. Yoram Cohen
Therefore, the total derivative of the system’s mass is zero (1.16)
Another way of stating eq. 1.16 is to set the right-hand-side of equation 1.15 to zero (1.17) where mass =
. Since we are dealing with an arbitrary volume, the integrand on the R.H.S of
Eq. 1.15 must equal zero. Therefore,
(1.18)
We now apply the argument presented in section 1.4 (eq. 1.6) and thus eq. 1.19a can be converted to the following differential form of the conservation of mass equation
(1.19)
For an incompressible fluid the density is invariant with respect to position and time, i.e., (1.20) which leads to the alternative form of the continuity equation for an incompressible fluid (1.21)
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1.2
Author: Dr. Yoram Cohen
Eulerian and Lagrangian Coordinates Eulerian coordinate - the independent variables are x, y, z and t or xi, t (i=1...3). This is a
fixed coordinate system. The basic conservation equations are in the Eulerian frame, R = R (xi,t). In the Lagrangian frame attention is fixed on a particular mass of fluid as it flows, R = R (xoi,t), where the coordinate xio specifies which fluid element is being considered.
1.3
Material Derivative Consider a variable " such that (1.2)
then the total differential of " can be expressed as
(1.3) division by a time differential *t leads to the following expression (1.4) After taking the limit *t 6 0 is obtained for the material derivative
(1.5)
in which vi is the fluid velocity in direction i. The material derivative (Lagrangian time derivative) represents the total change in " as seen by an observer who is moving with a particular fluid element In the Lagrangian frame we observe the particle for a time *t as it flows. The position of the particle changes by *xi while " changes by *".
-1-
Source: Mass Transfer Fundamentals
1.4
Author: Dr. Yoram Cohen
Control Volume Consider an arbitrary control volume,
(1.6) in which " is a fluid property, V is an arbitrary volume and L is a differential operator operating on
". The above integral equation can be satisfied for any arbitrary control volume only if L" = 0. Thus, as we will see later, Eq. (1.6) can be used to convert an integro-differential equation to a differential equation.
1.5
Reynolds Transport Theorem We begin by considering an arbitrary volume element, V, as it moves and deforms throug
a fluid domain. Fluid enters the volume element only through the open faces shown in the Figure
Figure 1.5.1 Arbitrary Fluid Volume Element As the volume element moves, volume-specific properties (e.g., density, enthalpy/volume) can change spatially and temporally. In order to describe the change of any specific volume-specific property, we can express the total time derivative of the selected property (integrated over the
-2-
Source: Mass Transfer Fundamentals
Author: Dr. Yoram Cohen
volume element) using a finite difference approximation over an infinitesimal time interval, *t. Accordingly, the change of a volume specific property, ", with time can be expressed as
(1.7)
Equation 1.7 can be written in a more compact form
(1.8) The volume can change as the control volume moves in space during a specified time period. Thus, the differential control volume is determined by the fluid inflow into the element (i.e., the velocity perpendicular to the face of the element) velocity perpendicular to the area of the element (1.9) where ds is the surface area, v is the velocity and n is the unit normal. Using Eq. 1.9, the balance equation 1.8 can be written in the following form (1.10a)
(1.10b) In order to reduce the above equation to a more convenient form we make use of the "divergence theorem" (or Gauss' theorem): (1.11) -3-
Source: Mass Transfer Fundamentals
Author: Dr. Yoram Cohen
in which a is any vector and n is the unit normal. Therefore, the first integral term on the R.H.S of Eq. 1.10b can be written as
(1.12) and Eq. 1.10b becomes (1.13)
Another convenient form for Eq. 1.13 is in terms of index notations,
(1.14) Equations 1.13 and 1.14 are two alternative forms of the Reynolds' Transport Theorem. In Eqs. 1.13 and 1.14 the LHS represents the Lagrangian Derivative and the RHS represents the Eulerian Derivatives.
1.6
Conservation of Mass The equation for conservation of mass can be derive by considering the density of the
medium as the property of interest. The density is substituted for " in the Reynolds transport equation ( eq. 1.13) resulting in the following relation
(1.15)
Conservation of mass requires (excluding nuclear reactions) that the mass of the system remains unchanged, assuming that there are no nuclear reactions in the domain of interest.
-4-
Source: Mass Transfer Fundamentals
Author: Dr. Yoram Cohen
Therefore, the total derivative of the system’s mass is zero (1.16)
Another way of stating eq. 1.16 is to set the right-hand-side of equation 1.15 to zero (1.17) where mass =
. Since we are dealing with an arbitrary volume, the integrand on the R.H.S of
Eq. 1.15 must equal zero. Therefore,
(1.18)
We now apply the argument presented in section 1.4 (eq. 1.6) and thus eq. 1.19a can be converted to the following differential form of the conservation of mass equation
(1.19)
For an incompressible fluid the density is invariant with respect to position and time, i.e., (1.20) which leads to the alternative form of the continuity equation for an incompressible fluid (1.21)
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