CHE150B PSET1 (Incomplete)

# CHE150B PSET1 (Incomplete) - -D AB ∇ c A-D BA ∇(1-c A =...

This preview shows pages 1–2. Sign up to view the full content.

CHEMICAL ENGINEERING 150B PROBLEM SET #1 ARJAN SINGH PUNIANI Problem 1 (a) D AB = D BA . Proof. Assume D AB 6 = D BA . Start with (1) N Az = - cD AB dy A dz + y A ( N Az + N Bz ) Keeping in mind that (2) N Bz = - cD BA dy B dz + y B ( N Bz + N Az ) Since we are dealing with a binary mixture, we can extract N Bz by substituting 1 - y B into the N Az equation above. Thus: N Az = - cD AB d (1 - y B ) dz + (1 - y B )( N Az + N Bz ) Simplifying to N Az = - cD AB - dy B dz + ( N Az + N Bz ) - y B ( N Az + N Bz ) We can eliminate a pair of N Az s, leaving 0 = cD AB dy B dz + N Bz - y B ( N Az + N Bz ) (3) N Bz = - cD AB dy B dz + y B ( N Az + N Bz ) Equating (2) and (3), we find that: - cD BA dy B dz + y B ( N Bz + N Az ) = - cD AB dy B dz + y B ( N Az + N Bz ) Eliminating y B ( N Bz + N Az ) (by associativity), - cD BA dy B dz + cD AB dy B dz = 0 ⇒ - c dy B dz ( D BA - D AB ) = 0 Date : 03 Sept 2010. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 ARJAN SINGH PUNIANI Which means D BA = D AB The contradiction completes the proof. Note this result holds true for the general case, where the gradient operator, ~ , acts on c rather than d dz . (b) J A + J B = 0 . Proof. Rewrite above as (4) - D AB c A - D BA c B = 0 Since c A + c B = 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: -D AB ∇ c A-D BA ∇ (1-c A ) = 0 ⇒ -D AB ∇ c A + D BA ∇ c A = 0 From ( a ) above, D AB = D BA so-D AB ∇ c A + D AB ∇ c A vanishes identically. ± (c) N A + N B = c V. Proof.-cD AB ~ ∇ y A + y A ( N A + N B )-cD BA ~ ∇ y B + y B ( N A + N B ) Substituting 1-y A for y B yields-cD AB ~ ∇ y A + y A ( N A + N B )-cD BA ~ ∇ (1-y A ) + (1-y A )( N A + N B ) Cancellation ensues between-cD AB ~ ∇ y A and y A ( N A + N B ), leaving ( N A + N B ) Where N A = c A v A and N B = c B v B , or c A v A + c B v B Which can be compacted to n X i =1 c i v i And, by deﬁnition, means that c A v A + c B v B = N A + N B = c V ±...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern