When liquid flows along a solid surface (see Fig. 4.2) a shearing stress is set up (friction power/surface), which is expressed by
R = for a plate (4.34)
FIGURE 4.1 Tube flow velocity profile.
T = feja = for a tube, (4.35)
Vm is mean velocity (velocity is zero at the surface)
F is a dimensionless friction factor
Ј, = 4f; this is the Blasius friction factor
In some literature A or /3 is used instead of Ј; it is essential to use because A is used for thermal conductivity and /3 is used for cubic expansion of air or an angle.
In a tube, the pressure is constant in all radial directions perpendicular to the axis; it varies only in the flow direction x. The power balance of the element dx, denoting the sectional surface area as A and the periphery as C, gives
-rCdx-&dxA = 0 and = =
Dx dx A I ‘ 2 A
Denoting the hydraulic diameter as dh = 4A/C, we have
And the friction resistance head corresponding to the pressure difference is
Hf=^- = lTTl — (4-37’l
; Pg sdh2g y
The factor Ј depends on the Reynolds number, which is a dimensionless variable that denotes the nature of flow:
Re = , (4.38)
V t] TTd 17
Vm is the mean velocity
D is the characteristic length of a surface; in the case of flow in a tube it is the tube diameter (note that d may be expressed by L in the case of a plate) v is the kinematic viscosity 77 is the dynamic viscosity qm in the mass flow
Viscosity is defined by means of the equation
Where dv/dy is the velocity gradient, t is the shearing stress between two flow layers, and rf is the dynamic viscosity.
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