Page 207 - Modelling in Transport Phenomena A Conceptual Approach

7.3. CONSERVATION OF TOTAL MASS 187

Analysis

System: Fluid in the tank
The inventory rate equation for total mass, Eq. (7.3-1), reduces to

where (v,) is the average velocity through the orifice, i.e., the volumetric flow rate
divided by the cross-sectional area; A, and A are the cross-sectional areas of orifice
and the tank, respectively. Since p and A are constant, Eq. (1) becomes

In order to proceed further, (v,) must be related to h.
For flow in a pipe of uniform cross-sectional area A, the pressure drop across
an orifice is given by

(3)

where /3 is the ratio of the orifice diameter to the pipe diameter, IAPI is the pres-
sure drop across the orifice, and C, is the orifice coeficient. The value of C,
is generally determined from experiments and given as a function of p and the
Reynolds number, Re,, defined by

For /3 < 0.25, the term d-is almost unity. On the other hand, when Re, >
20,000, experimental measurements show that C, N 0.61. Hence, Eq. (3) reduces