Question

Some discarded solid chemical waste dissolves slowly in a large drain pipe in which the water is stagnant. On a particular day, the dissolved chemical has a concentration of 0.16 M near the solid and is essentiallly 0 at a location 13.6 m further along the pipe. The transfer rate (moles per time) of tthe chemical through the water in the pipe between those two points is 7.3 gmol/min.

A. What equation describes this kind of transfer?

B. Several days later, the chemical concentration near the solid has decreased to 0.105 M and is essentially 0 at a location 9.9 m away. What will be the transfer rate between the point near the solid and point 9.9 m away on this later day?

C.On the next day, the heavy rains cause a current of waater to flow through the drain pipe where the dissolving solid is located. The solid now dissolves twice as fast as on the previous day. If the concentration are still 0.105 M near the solid and 0 at the more remote locations and if the cross-sectional area for transfer is 0.3 m^2 , what is the value of the mass-transfer coefficient at this time?

Answer 1

The mass transfer coefficient, is a parameter that is used to describe the ratio between the actual mass (or molar) flux of a species into or out of a flowing fluid and the driving force that causes that flux. For example, if a liquid flows over a solid surface that is dissolving in the liquid, one might write N kc c k c A c As A c A = − =∆ ( , ,∞ ) where A s, c is the concentration of the solute A in the liquid in contact with the solid surface, which is assumed to be the equilibrium concentration or solubility, and A, c ∞ is the concentration of A in the liquid far from the solid surface. Here, c k is defined as the mass transfer coefficient in this situation, based on a concentration driving force. It is possible to define a mass transfer coefficient in the same situation using a mole fraction driving force. N kx x k x A x As A x A = − =∆ ( , ,∞ ) Given the geometry, the fluid and flow conditions, and the prevailing thermodynamic conditions, the molar flux must be the same, regardless of the type of driving force used. Thus, in this example, the two mass transfer coefficients are related to each other through x Ac A kxkc ∆ =∆ We define the mole fraction / A A x cc = , where c is the total molar concentration of the mixture. Thus, / A A ∆ =∆ x cc . Substituting in the above result yields the connection between the two mass transfer coefficients. x c k ck = Mass transfer coefficients depend on the relevant physical properties of the fluid, the geometry used along with relevant dimensions, and the average velocity of the fluid if we are considering flow in an enclosed conduit, or the approach velocity if the flow is over an object. Dimensional analysis can be used to express this dependence in dimensionless form. The dimensionless version of the mass transfer coefficient is the Sherwood number Sh . c AB k D Sh D = 2 where D is a characteristic length scale in the problem, such as the diameter of a tube through which fluid flows, or the diameter of a sphere or cylinder over which fluid flows. In terms of the mass transfer coefficient x k , we define the Sherwood number as Sh k D cD = x /( AB ).

Kx=CKc

7.3=0.16xKc

Kc=7.3/0.16=45.625

Sherwood No=Kcx13.6=620.5

N_A =K_c(CA_x-C_A)= 45.625x(0.16-0)=45.625x0.16=7.3

2)

N_A =K_c(CA_x-C_A)= 45.625x(0.105-0)=45.625x0.105=4.79

3)

N_A =K_c(CA_x-C_A)= 45.625x(0.105-0)=45.625x0.105=4.79

Sherwood no Sh= K_c D/ D_AB=45.625D/diameter