User:Egm6936.f09/Particle time scale

Egm6936.f09 16:41, 21 October 2009 (UTC)

In a particle-laden flow, the particle phase is also known as the dispersed phase, whereas the carier fluid is called the carier phase. The dispersed phase can be modeled either as a system of discrete particles or as a continuous fluid. The carier phase here is a continuous fluid medium. In point-particle methods, each particle is modeled as a material point, and the particle phase is a system of discrete point particles.

The particle time scale $$\displaystyle \tau_p$$, also known as the momentum (velocity) response time, is the time for a particle to respond to a change in the flow velocity of the carier fluid, and is given by (Crowe et al. (1997), p.23)


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$$  \displaystyle \tau_p :=  \frac {     \rho_p (D_p)^2 }  {      18 \mu_c } $$     (1)
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where $$\displaystyle \rho_p$$ is the density of the particle material, $$\displaystyle D_p$$ the particle diameter, $$\displaystyle \mu_c$$ the viscosity of the carier phase.

Note: The subscript $$\displaystyle c$$ in $$\displaystyle \mu_c$$ may seem redundant for now (since there is no $$\displaystyle \mu_p$$ ), but is kept to quickly discern quantities related to the carier phase versus quantities related to the particle phase.

= Dimension check =

Using Newton's law and Hooke's law:

\displaystyle [\rho_p] =   \frac {\rm mass} {\rm volume} =   \frac {     F L^{-1} T^2 }  {      L^3 }  \, \quad [(D_p)^2] =  L^2 \, \quad [\mu_c] =  F L^{-2} T $$

Recall Hooke's law

\displaystyle \sigma_{xy} =   \mu_c \epsilon_{xy} \, \quad [\sigma_{xy}] =  F L^{-2} \, \quad [\epsilon_{xy}] =  T^{-1} $$ where $$\displaystyle \sigma_{xy}$$ is the shear stress, $$\displaystyle \epsilon_{xy}$$ the shear strain rate.

Thus

\displaystyle [\tau_p] =  \frac {     (F L^{-4} T^2) (L^2) }  {      F L^{-2} T   } =  T $$

Physically, Eq.(1) says that $$\displaystyle \tau_p$$ would be long if (a) the particle is heavy (high material density $$\displaystyle \rho_p$$ ), (b) the particle is large (high value of particle diameter $$\displaystyle D_p$$ ), (c) the carier fluid is less viscous (in other words, the more viscous the fluid, the shorter the particle response time, since a change in velocity in a more viscous fluid would move the particle along quicker than a less viscous fluid).

= Derivation =

Kinetic friction force
The distributed friction force on the surface of a sphere submerged in a creeping flow has a resultant called the kinetic friction force $$\displaystyle F_k$$, having the following expression (Bird et al. (2002), p.186, Eq.(6.3-14)
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$$  \displaystyle F_k =  A    \cdot K   \cdot f  = \left[ \pi (R_p)^2 \right] \left[ \frac{1}{2} \rho_c ( v_\infty )^2 \right] f  = \frac {1}  {2}   f   \frac {\pi (D_p)^2} {4}  \rho_c ( v_\infty )^2 $$     (3)
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where $$\displaystyle A = \pi (R_p)^2$$, with $$\displaystyle R_p = D_p / 2$$, is the characteristic area of the spherical particle, $$\displaystyle K = \frac{1}{2} \rho_c ( v_\infty )^2 $$ the kinetic energy per unit fluid volume at infinity, with $$\displaystyle \rho_c$$ being the carier density and $$\displaystyle v_\infty$$ the terminal velocity, and $$\displaystyle f$$ is the Fanning friction factor (Bird et al. (2002), p.179, Eq.(6.1-5)). For a particle in a flow, the Fanning friction factor is the same as the drag coefficient

$$\displaystyle C_D$$ of the particle.

Since the product $$\displaystyle A \cdot K$$ has dimension of force, i.e.,

\displaystyle [A \cdot K]  = [A] \cdot [K] =  L^2 \cdot \frac {FL} {L^3} =  F   \ , $$ it follows that $$\displaystyle f \equiv C_D$$ is dimensionless, as expected.

Note that Eq.(3) is the definition of the dimensionless quantity $$\displaystyle f \equiv C_D$$, and is not a fluid mechanics law (Bird et al. (2002), p.178, Eq.(6.1-5)).

The Fanning friction factor $$\displaystyle f \equiv C_D$$ can be measured by dropping a sphere in a fluid, and measure the terminal velocity $$\displaystyle v_\infty$$. By considering the equilibrium of the forces that exert on the sphere, namely the kinetic friction force $$\displaystyle F_k$$, the weight of the sphere $$\displaystyle F_g$$, and the buoyant force $$\displaystyle F_b$$, one can deduce the following expression for $$\displaystyle f \equiv C_D$$

\displaystyle F_k =  F_g -  F_b \Longrightarrow \left[ \pi (R_p)^2 \right] \left[ \frac{1}{2} \rho_c ( v_\infty )^2 \right] f  = \frac {4 \pi (R_p)^3} {3}  \left[ \rho_p - \rho_c \right] \Longrightarrow f  \equiv C_D =  \frac {4}  {3}   \frac {g D_p} {( v_\infty )^2} \frac {\rho_p - \rho_c} {\rho_c} $$

Equation of motion
To derive Eq.(1), let $$\displaystyle v_\infty = v_c - v_p$$, where $$\displaystyle v_c$$ is the velocity of the carier phase and $$\displaystyle v_p$$ the velocity of the particle, then consider only the kinetic friction force $$\displaystyle F_k$$ acting on the particle. Using Eq.(3) in Newton's law, the equation of motion of a single particle in a creeping fluid is given by
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$$  \displaystyle m_p \frac {d v_p} {d t}  = \frac{1}{2} C_D \frac{\pi (D_p)^2}{4} \rho_c \left(     v_c - v_p   \right) \left| v_c - v_p \right| \, \quad m_p =  \rho_p \frac {4 \pi R_p} {3}  =   \rho_p \frac {\pi (D_p)^3} {6} $$     (4) where $$\displaystyle m_p$$ is the mass of a particle.
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Particle Reynolds number
The right hand side of Eq.(4) is the kinetic friction force $$\displaystyle F_k$$ of creeping flow around a sphere (Bird et al. (2002), p.186, Eq.(6.3-14), in which the drag coefficient $$\displaystyle C_D$$ is often denoted by $$\displaystyle f$$, called the Fanning friction factor (Bird et al. (2002), p.179, Eq.(6.1-5))

The particle Reynolds number

is defined as (Brodkey & Hershey (1988), p.589, Eq.(12.68))
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$$  \displaystyle Re_p :=  \frac {\rho_c D_p \left| v_c - v_p \right| } {\mu_c} $$     (5) where the fluid inertial force $$\displaystyle F_i$$ is
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\displaystyle F_i =  \rho_c (v_c - v_p)^2 $$ and the fluid viscous force is

\displaystyle F_v =  \mu_c \frac {\left| v_c - v_p \right|} {D_p} $$ The Reynolds number is the ratio of inertial force over viscous force:

\displaystyle Re_p =  \frac {F_i} {F_v} $$

Drag coefficient
For low particle Reynolds number (Stokes flow), i.e., with $$\displaystyle Re_p < 800$$, we have the following approximation (Clift et al. (2005), p.111)


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$$  \displaystyle C_D =   \frac{24}{Re_p} \phi(Re_p) \, \quad \phi (Re_p) =  1 + 0.15 (Re_p)^{0.687} \ . $$     (7a)
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See also Kee & Chhabra (2002), p.23, where it is mentioned that Oseen (1910, 1913) derived the expression
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$$  \displaystyle C_D =   \frac{24}{Re_p} \phi( Re_p ) =  1 + \frac{3}{8} Re_p \ . $$     (7b)
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Both Eqs.(7a) and (7b) lead to the the following approximation for the dimensionless ratio
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$$  \displaystyle \frac {C_D Re_p} {24}  \approx 1  \ . $$      (7c)
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Particle response time
Using Eq.(5) in Eq.(4), the particle acceleration can be written as
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$$  \displaystyle \frac {d v_p} {d t}  = \frac {1}  {\rho_p} \frac {6}  {\pi (D_p)^3} \cdot \frac{1}{2} C_D \frac{\pi (D_p)^2}{4} \rho_c \left(     v_c - v_p   \right) \left| v_c - v_p \right| =  \frac {18 \mu_c} {\rho_p (D_p)^2} \frac {C_D Re_p} {24}  (v_c - v_p) $$     (10)
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It follows from Eq.(7c) that the coefficient $$\displaystyle (18 \mu_c) / [ \rho_p (D_p)^2 ]$$ has the dimension of $$\displaystyle T^{-1}$$ (time inverse). The particle time scale $$\displaystyle \tau_p$$ is therefore defined as in Eq.(1).

= Meaning =

The acceleration of the particle in Eq.(4) can then be re-written as
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$$  \displaystyle \frac {d v_p} {d t}  = \frac {1}  {\tau_p} (v_c - v_p) $$     (15)
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At $$\displaystyle t=0$$, we have
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$$  \displaystyle v_p (0) = 0 $$     (16)
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Hence
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$$  \displaystyle v_p (t) =  v_c \left[ 1 - exp (- t / \tau_p ) \right] $$     (17)
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Thus, at the time equal to the particle time scale $$\displaystyle t = \tau_p$$, the particle velocity is equal to
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$$  \displaystyle v_p (\tau_p) =  v_c \left[ 1 - exp (- 1) \right] =  0.63   \cdot v_c $$     (18) or 63% of the carier flow velocity $$\displaystyle v_c$$. So the particle response time is the time it takes for the particle velocity to reach 63% of the carier velocity.
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= References =