0.5 Equations of motion and energy in cartesian coordinates (Page 12/12)

Transport phenomena Page 12 / 12

Friction factors
$f_{S P} = \frac{τ_{w}}{ρ U_{m}^{2}}$	Stanton-Pannel
$f_{F} = \frac{2 τ_{w}}{ρ U_{m}^{2}}$	Fanning
$f_{D W} = \frac{8 τ_{w}}{ρ U_{m}^{2}}$	Darcy-Weisbach
$f_{M o o d y =} \frac{2 D Δ P / L}{ρ U_{m}^{2}} = 4 f_{F} = f_{D W}$	Moody
$f_{d r a g} = \frac{F_{f r i c t i o n} + F_{d r a g}}{A_{p} (1 / 2 ρ U_{\infty}^{2})}$	drag coefficient
${\bar{C}}_{t} = \frac{F_{f r i c t i o n} + F_{d r a g}}{A_{p} ρ U_{\infty}^{2}}$	drag coefficient

Nomenclature of S. W. Churchill

$\begin{matrix} A_{p} projected area, m^{2} \\ {\bar{C}}_{f} = \frac{F_{f}}{A_{p} ρ u_{\infty}^{2}}, mean drag coefficient due to friction \\ {\bar{C}}_{p} = \frac{F_{p}}{A_{p} ρ u_{\infty}^{2}}, mean drag coefficient due to pressure \\ {\bar{C}}_{t} = \frac{F_{f} + F_{p}}{A_{p} ρ u_{\infty}^{2}}, total mean drag coefficient \\ F_{f} drag force due to friction, N \\ F_{p} drag force due to pressure, N \end{matrix}$

Bernoulli theorems

When the viscous effects are negligible compared with the inertial forces (i.e., large Reynolds number) there are a number of generalizations that can be made about the flow. These are described by the Bernoulli theorems. The fluid is assumed to be inviscid and have zero thermal conduction so that the flow is also barotropic (density a single-valued function of pressure). The first of the Bernoulli theorems is derived for flow that may be rotational. A special case is for motions relative to a rotating coordinate system where Coriolis forces arise. For irrotational flow, the Bernoulli theorem is a statement of the conservation kinetic energy, potential energy, and the expansion energy. A macroscopic energy balance can be made that includes the effects of viscous dissipation and the work done by the system.

Steady, barotropic flow of an inviscid, nonconducting fluid with conservative body forces . The equations of motion for a Newtonian fluid is

ρ \frac{D v}{D t} = ρ f - \nabla p + (λ + μ) \nabla (\nabla • v) + μ \nabla^{2} v

The assumptions of steady, inviscid flow simplify the equations to

ρ (v • \nabla) v = ρ f - \nabla p

The assumptions of barotropic flow with conservative body forces allow,

\begin{matrix} f = - \nabla Ω, Φ (p) = \int_{}^{p} \frac{d p}{ρ} \\ (v • \nabla) v = - \nabla (Ω + Φ (p)) \end{matrix}

and by virtue of the identity

(v • \nabla) v = \nabla (\frac{1}{2} v^{2}) + w \times v

the equations of motion can be written

\begin{matrix} \nabla (Ω + Φ (p) + \frac{1}{2} v^{2}) = v \times w \\ \nabla H = v \times w \\ H \equiv Ω + Φ (p) + \frac{1}{2} v^{2} \end{matrix}

If the body force is gravitational then $Ω = ρ g z$ .

Let $H$ denote the function of which the gradient occurs on the left-hand side of this equation. $\nabla H$ is a vector normal to the surfaces of constant $H$ . However, $v \times w$ is a vector perpendicular to both $v$ and $w$ so that these vectors are tangent to the surface. However, $v$ and $w$ are tangent to the streamlines and vortex lines respectively so that these must lie in a surface of constant $H$ . It follows that $H$ is constant along the streamlines and vortex lines. The surfaces of constant $H$ which are crossed with this network of stream and vortex lines are known as Lamb surfaces and are illustrated in Fig. 6.2.

Coriolis force. Suppose that the motion is steady relative to a steadily rotating axis with an angular velocity, $ω$ . Batchelor (1967) derives the equations of motion in this rotating frame and shows that we must now include the potential from the centrifugal force and the addition of a Coriolis force term.

\begin{matrix} \nabla H = v \times (w + 2 ω) \\ H \equiv Ω + Φ (p) + \frac{1}{2} v^{2} - \frac{{(ω \times x)}^{2}}{2} \end{matrix}

Irrotational flow. If the flow is also irrotational, then $w = 0$ and hence the energy function

H \equiv Ω + Φ (p) + \frac{1}{2} v^{2}

is constant everywhere.

Ideal gas. For an ideal gas we have

p = (c_{p} - c_{v}) ρ T

and if the heat capacities are assumed constant an isentropic change of state results in the following expression for $H$ .

H_{ideal gas} = Ω + c_{p} T + \frac{1}{2} v^{2}

The gas is hotter at places on a streamline where the speed is smaller or has a lower potential energy. This may represent the heating of air at a stagnation point, cooling of ascending air, or heating of descending air.

The transformation from the function of pressure $Φ (p)$ to heat capacity and temperature in the above equation for the isoentropic expansion of an ideal gas with constant heat capacity is proven as follows. The total differential of entropy in terms of pressure and temperature is as follows.

d S = {(\frac{\partial S}{\partial p})}_{T} d p + {(\frac{\partial S}{\partial T})}_{p} d T

In an isoentropic expansion, the change of entropy is zero and this gives us a relation between the differential of pressure and differential of temperature.

{(\frac{\partial S}{\partial p})}_{T} d p = - {(\frac{\partial S}{\partial T})}_{p} d T, for d S = 0

The coefficient on the left-hand side can be determined from the Maxwell relations and the ideal gas EOS.

\begin{matrix} {(\frac{\partial S}{\partial p})}_{T} & = & - {(\frac{\partial V}{\partial T})}_{p}, Maxwell relation \\ = & - \frac{(c_{p} - c_{v})}{p} = - \frac{1}{ρ T}, ideal gas \end{matrix}

The coefficient of the right hand side can be determined from the definition of the heat capacity at constant pressure.

\begin{matrix} c_{p} & = & {(\frac{\partial^{'} q}{\partial T})}_{p}, reversible process \\ = & T {(\frac{\partial S}{\partial T})}_{p} \\ ∴ {(\frac{\partial S}{\partial T})}_{p} = \frac{c_{p}}{T} \end{matrix}

Substituting these relations into the equation gives us the following.

\begin{matrix} \frac{d p}{ρ} & = & c_{p} d T \\ \int_{}^{p} \frac{d p}{ρ} & = & \int_{}^{T} c_{p} d T \\ = & c_{p} T, if c_{p} is constant \\ Q . E . D . \end{matrix}

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Source: OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1

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