Page 94 - Modelling in Transport Phenomena A Conceptual Approach

74 CHAPTER 4. EVALUATION OF TRANSFER COEFFICIENTS

4.3 FLOW PAST A SINGLE SPHERE

Consider a single sphere immersed in an infinite fluid. We may consider two cases
which are exactly equivalent: (i) the sphere is stagnant, the fluid flows over the
sphere, (ii) the fluid is stagnant, the sphere moves through the fluid.
According to Newton’s second law of motion, the balance of forces acting on a
single spherical particle of diameter Dp, falling in a stagnant fluid with a constant
terminal velocity ut, is expressed in the form
Gravitational force = Buoyancy + Drag force (4.3-1)

(4.3-2)

where pp and p represent the densities of the particle and fluid, respectively. In
the literature, the friction factor f is also called the drag coeficient and denoted
by C,. Simplification of Q. (4.3-2) gives

(4.3-3)

Equation (4.3-3) can be rearranged in dimensionless form as

(4.3-4)

where the Reynolds number, Rep, and the Archimedes number, Ar, are defined by

(4.3-5)

(4.3-6)
Engineering problems associated with the motion of spherical particles in fluids are
classified as follows:
0 Calculate the terminal velocity, vt; given the viscosity of fluid, p, and the
particle diameter, Dp.
0 Calculate the particle diameter, Dp; given the viscosity of the fluid, p, and
the terminal velocity, vt.
0 Calculate the fluid viscosity, p; given the particle diameter, Dp, and the
terminal velocity, ut.
The difficulty in these problems arises from the fact that the friction factor f in Eq.
(4.3-4) is a complex function of the Reynolds number and the Reynolds number
cannot be determined a priori.