Equation of Continuity. differential control volume:
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Transcript of Equation of Continuity. differential control volume:
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Equation of Continuity
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Equation of Continuity
differential control volume:
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Differential Mass Balance
Rate of Rate of Rate of
accumulation mass in mass out
mass balance:
Rate of
mass in
x y zx zyv y z v x z v x y
Rate of
mass out
x y zx x z zy yv y z v x z v x y
Rate of mass
accumulationx y z
t
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Differential Equation of Continuity
yx zvv v
t x y z
v
divergence of mass velocity vector (v)
Partial differentiation:
yx zx y z
vv vv v v
t x y z x y z
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Differential Equation of Continuity
Rearranging:
yx zx y z
vv vv v v
t x y z x y z
substantial time derivative
yx zvv vD
Dt x y zv
If fluid is incompressible: 0 v
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Equation of Continuity
π· ππ·π‘
=βπ (π» βπ )ππππ‘
=β(π» β Ο π) or
Conservation of mass for pure liquid flow
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Equation of Continuity
Applying the conservation of mass to the volume element
* May also be expressed in terms of moles
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Equation of Continuity
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Equation of Continuity
( πππ‘π πππππ π π΄ππ)=ππ΄π₯β¨ π₯β π¦ β π§ ( πππ‘πππ
πππ π π΄ππ’π‘)=ππ΄π₯β¨ π₯+β π₯β π¦ β π§
( πππ‘ππππππππππ‘πππππ πππ π π΄)=π π΄ βπ₯ β π¦ β π§
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Equation of Continuity
ππ΄π₯β¨π₯β π¦ β π§βππ΄π₯β¨π₯ +β π₯β π¦ β π§+π π΄β π₯β π¦ β π§=π π π΄
ππ‘βπ₯ β π¦ β π§
Dividing by and letting approach zero,
ππ π΄
ππ‘+( πππ΄π₯
π π₯+
πππ΄ π¦
π π¦+
πππ΄π§
π π§ )=π π΄
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Equation of Continuity
ππ π΄
ππ‘+( πππ΄π₯
π π₯+
πππ΄ π¦
π π¦+
πππ΄π§
π π§ )=π π΄
ππ π΄
ππ‘+( π» βππ΄ )=π π΄
In vector notation,
But form the Table 7.5-1 (Geankoplis)
ππ΄= ππ΄+ππ΄ π£ ππ΄=β π π·π΄π΅ π π€π΄/ ππ§and
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Equation of Continuity
ππ π΄
ππ‘+( π» βππ΄ )=π π΄
ππ΄= ππ΄+ππ΄ π£ ππ΄=β π π·π΄π΅ π π€π΄/ ππ§
Substituting
ππ π΄
ππ‘+( π» βπ π΄ π£ )β ( π» βπ π· π΄π΅ π»π€ π΄ )=π π΄
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Equation of Continuity
ππ π΄
ππ‘+( π» βππ΄ )=π π΄
ππ π΄
ππ‘+( π ππ΄π₯
ππ₯+
π π π΄ π¦
π π¦+
π π π΄ π§
π π§ )=π π΄
Dividing both sides by MWA
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Equation of Continuity
Recall: 1. Fickβs Law
π½ π΄β=βπ·π΄π΅
π ππ΄
ππ§2. Total molar flux of A
π π΄= π½ π΄β +ππ΄ π£π
π π΄=βππ·π΄π΅
π π₯π΄
ππ§+π₯π΄(π π΄+ππ΅)
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Equation of Continuity
ππ π΄
ππ‘+( π ππ΄π₯
ππ₯+
π π π΄ π¦
π π¦+
π π π΄ π§
π π§ )=π π΄
Substituting NA and Fickβs lawand writing for all 3 directions,
ππ π΄
ππ‘+( π» βπ π΄ π£π )β ( π» βπ π· π΄π΅ π» π₯π΄ )=π π΄
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Equation of Continuity
ππ π΄
ππ‘+( π» βπ π΄ π£π )β ( π» βπ π· π΄π΅ π» π₯π΄ )=π π΄
ππ π΄
ππ‘+( π» βπ π΄ π£ )β ( π» βπ π· π΄π΅ π»π€ π΄ )=π π΄
(πππ‘πππ
πππππππ ππππππππ ππ π΄πππ
π’πππ‘ π£πππ’ππ)(
πππ‘ πππ‘ππππππππ‘πππ
πππππππ πππ΄πππ π’πππ‘π£πππ’ππππ¦ππππ£πππ‘πππ
)(πππ‘ πππ‘ππππππππ‘πππ
πππππππ πππ΄πππ π’πππ‘π£πππ’ππππ¦πππππ’π πππ
)(πππ‘πππ
πππππ’ππ‘πππππ πππππ πππ΄πππ π’πππ‘π£πππ’ππ
ππ¦ πππππ‘πππ)
Two equivalent forms of equation of continuity
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Equation of Continuity
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Equation of Continuity
ππ π΄
ππ‘+( π» βπ π΄ π£π )β ( π» βπ π· π΄π΅ π» π₯π΄ )=π π΄
Special cases of the equation of continuity
1. Equation for constant c and DAB,
At constant P and T, c= P/RT for gases, and substituting
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Equation of Continuity
ππ π΄
ππ‘+( π» βπ π΄ π£π )β ( π» βπ π· π΄π΅ π» π₯π΄ )=π π΄
Special cases of the equation of continuity
2. Equimolar counterdiffusion of gases,
At constant P , with no reaction, c = constant, vM = 0, DAB = constant and RA=0
Fickβs 2nd Law of diffusion
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Equation of Continuity
ππ π΄
ππ‘+( π» βπ π΄ π£π )β ( π» βπ π· π΄π΅ π» π₯π΄ )=π π΄
Special cases of the equation of continuity
3. For constant Ο and DAB (liquids),
ππ π΄
ππ‘+( π» βππ΄ )=π π΄
Starting with the vector notation of the mass balance
We substitute and
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Equation of Continuity
Example 1.
Estimate the effect of chemical reaction on the rate of gas absorption in an agitated tank. Consider a system in which the dissolved gas A undergoes an irreversible first order reaction with the liquid B; that is A disappears within the liquid phase at a rate proportional to the local concentration of A. What assumptions can be made?
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Equation of Continuity
1. Gas A dissolves in liquid B and diffuses into the liquid phase
2. An irreversible 1st order homogeneous reaction takes place
A + B AB
Assumption: AB is negligible in the solution (pseudobinary assumption)
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Equation of Continuity
Expanding the equation and taking c inside the space derivative,
β ( π·π΄π΅ π»2 π π΄ )=π π΄
Assuming steady-state,
ππ π΄
ππ‘β ( π·π΄π΅ π»2 π π΄ )=π π΄
Assuming concentration of A is small, then total c is almost constant and
ππ π΄
ππ‘+( π» βπ π΄ π£π )β ( π» βπ π· π΄π΅ π» π₯π΄ )=π π΄
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Equation of Continuity
Assuming that diffusion is along the z-direction only,
β(π· π΄π΅
π2ππ΄
π π§2 )=π π΄
β ( π·π΄π΅ π»2 π π΄ )=π π΄
We can write that since A is disappearing by an irreversible, 1st order reaction
β(π· π΄π΅
π2π π΄
π π§2 )=βπ1β² β² β²π π΄
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Equation of Continuity
π· π΄π΅
π2 ππ΄
π π§2 βπ1β² β² β²ππ΄=0
Rearranging,
Looks familiar?How to solve this ODE?
π π΄
π π΄0
=cosh [β πβ² β² β² πΏ2
π· π΄π΅(1β π§
πΏ )]cosh (β πβ² β² β² πΏ2
π· π΄π΅
)
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Equation of Continuity
A hollow sphere with permeable solid walls has its inner and outer surfaces maintained at a constant concentration CA1 and CA0 respectively. Develop the expression for the concentration profile for a component A in the wall at steady-state conditions. What is the flux at each surface?
Example 2.