Transport phenomena (Continuity Equation)

Transport phenomena (Continuity Equation in 3 dimensions).
Derivation of Continuity Eq.

1. Transport Phenomena
CONTINUITY EQUATION IN 3
DIMENSIONS:

2. 2
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Chemical Engineering student @Mehran University
of Engineering & Technology Jamshoro, Pakistan.
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3. Transport Phenomena
3
CONTINUITY EQUATION IN 3
DIMENSIONS:
“Theequationbasedontheprincipleofconversationofmass.”
A
E
C
D
G
F
H
B
dx
dz
2kg
1kg
1kg
x-axis (u)
y-axis (v)
z-axis (w)

4. 4
▪Law of conservation of mass
CONTINUITY EQUATION IN 3
DIMENSIONS:
�??????����
??????����??????��
���??????��
=
�??????����
�??????��
??????�
−
�??????����
�??????��
���
Transport Phenomena

5. Transport Phenomena
5
CONTINUITY EQUATION IN 3
DIMENSIONS:
Directionofx,y,z.
uinletvelocitycomponentsinxdirection.
vinletvelocitycomponentsinydirection.
winletvelocitycomponentsinzdirection.
Massoffluidenteringtheface+vefaceABCD(inflow)per
second .
�
�
����??????��??????=
�
??????
�=??????×??????
�
�

??????×??????
�

??????��
�
??????×���??????������×�����??????��??????���??????����??????��
=??????�????????????×�×(��×��)

6. 6
MassoffluidleavingthefaceEFGH(outflow)persecond,
CONTINUITY EQUATION IN 3
DIMENSIONS:
??????�����+
??????
??????�
??????�������
Rateofincreaseinmassx-direction=massthroughABCD–massthroughEFGH
=??????�����−??????�����−
??????
??????�
??????�������
=−
??????
??????�
??????�������
=−
??????
??????�
??????������z
Transport Phenomena

7. 7
Similarly,
Rateofincreaseinmassiny-direction=−
??????
??????�
??????������z
Rateofincreaseinmassiny-direction=−
??????
??????�
??????������z
➢Totalrateofincreaseinmass=−
??????
??????�
??????�+
??????
??????�
??????�+
??????
??????�
??????�������
CONTINUITY EQUATION IN 3
DIMENSIONS:
Bythelawofconversationofmass,thereisnoaccumulationofmass
i.emassisneitherbecreatednordestroyedinthefluidelement.
Sonetincreaseofmassperunittimeinthefluidelementmustbe
equaltotherateofincreaseofmassoffluidintheelement.
...(1)
Transport Phenomena

8. 8
CONTINUITY EQUATION IN 3
DIMENSIONS:
Massoffluidintheelementis=??????������??????=
??????
??????
�=????????????
itsrateofincreasewithtime
=
??????
????????????
??????������
=
????????????
??????�
������
Equatingeq.(i)and(ii)
−
??????
??????�
??????�+
??????
??????�
??????�+
??????
??????�
??????�������=
????????????
??????�
������
…(2)
Transport Phenomena

9. 9
CONTINUITY EQUATION IN 3
DIMENSIONS:
????????????
??????�
=−
??????
??????�
??????�+
??????
??????�
??????�+
??????
??????�
??????�
Thisistheequationofcontinuity,whichdescribesthetimerateof
changeofthefluiddensityatafixedpointinspace.
Thisequationcanbewrittenmoreconciselybyusingvector
notationasfollows).(V
t



−=


Rate of increase
of mass per unit
volume
Net rate of addition of
mass per unit volume
by convection
Transport Phenomena

10. 
????????????
??????�
+
??????
??????�
??????�+
??????
??????�
??????�+
??????
??????�
??????�=0
▪Forsteadystateflow
????????????
????????????
=0andhenceaboveequationbecomes
as,
??????
??????�
??????�+
??????
??????�
??????�+
??????
??????�
??????�=0
▪Ifthefluidisincompressible,then??????isconstantandtheaboveeq.
becomesas,
??????�
??????�
+
??????�
??????�
+
??????�
??????�
=0
CONTINUITY EQUATION IN 3
DIMENSIONS:
Continuityequationinthree-dimensions.
Transport Phenomena

11. Thanks!
11