midterm1 2005 solutions - Chemical Engineering 140...

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Unformatted text preview: Chemical Engineering 140 September 26, 2005 Midterm #1 Solution (100) 1. V Conductivity probe C(t) SCL Ň Donor cell co A Given in problem statement: L Receptor cell A ≡ Area of membrane (SCL) V ≡ Volume of receptor cell L ≡ Thickness of membrane (SCL) co ≡ Concentration of salt in donor cell, CONSTANT Both tanks are WELL MIXED C(t = 0) = 0 (Upper chamber initially contains no salt) ( ) i. Use transient species mass balance on salt in upper chamber CV = Upper chamber volume = V Mass balance for species i = salt. No reaction, no outlet streams 0 0 d (nM ) dmi (m) nin M mi ,in mi ,out Gi dt dt Constant species molecular weight M allows us to reduce mass balance to mole balance d d (CV ) C (r )dV dt nin (Well mixed tank C is indep. of position) (*) dt V But, rate of moles in varies with the concentration difference across the SCL: PA ˆ co C (t ) (Relationship for flux N is given in part (i).) (**) nin NA L Combining (*) and (**), we obtain 0 d (CV ) dV dC PA co C (t ) (Expand by product rule; V constant) C V dt dt dt L V dC PA co C (t ) dt L Rearranging, we have a first order, linear ODE for C(t): dC PA PA C (t ) co dt VL VL ( ) (ii) Solve the ODE and use BC’s to get expression for concentration Defining an integrating factor and writing the general solution to this linear first order ODE, where is a constant of integration PA exp[ dt ] VL PAco 1 PA C (t ) dt , which simplifies to C (t ) c o exp t VL VL Applying the initial condition that C (t = 0) = 0 and simplifying gives PA C (t ) c o 1 exp t VL ( ) (iii) Express similar to Nicolas et al 1 C (t ) PA exp t co VL C (t ) PA ln 1 t co VL Equation 1 given by Nicolas is correct. ( ) (iv) Examine procedure for determining P: Plotting method by Nicolas et al is incorrect. Slope does not give P, as the slope has the wrong dimensions. Permeability P should be given in length2/time. Plotting ln(1-C/co)(-A/(LV)) versus t gives length 2 1 1 3 length length length 2 abscissa [=] time ordinate [=] slope [=] 1 length 2 time V, L, and A are known for the system. C(t) is measured by the conductivity meter. To obtain the proper value for P: Plot VL C (t ) ln 1 versus t on rectangular coordinates; slope = -P A co Or C (t ) PA Plot 1 versus t on semilog paper; slope = co VL ( ) (v) Conclusion: Although the species mass balance on the salt was correct, any claimed permeabilities P obtained for this product should be regarded as suspect, since P was found using an improper plotting method. ...
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This note was uploaded on 02/05/2010 for the course CHM ENG 140 taught by Professor Radke during the Fall '08 term at University of California, Berkeley.

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