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:
“The equation based on the principle of conversation of mass.”
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:
𝑟𝑎𝑡𝑒 𝑜𝑓
𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒
𝑜𝑓 𝑚𝑎𝑠𝑠
=
𝑟𝑎𝑡𝑒 𝑜𝑓
𝑚𝑎𝑠𝑠
𝑖𝑛
−
𝑟𝑎𝑡𝑒 𝑜𝑓
𝑚𝑎𝑠𝑠
𝑜𝑢𝑡
Transport Phenomena

5. Transport Phenomena
5
CONTINUITY EQUATION IN 3
DIMENSIONS:
Direction of x, y, z.
u  inlet velocity components in x direction.
v  inlet velocity components in y direction.
w  inlet velocity components in z direction.
Mass of fluid entering the face +ve face ABCD (inflow) per
second .
𝑚
𝑠
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝜌 =
𝑚
𝑉
 𝑚 = 𝜌 × 𝑉
𝑚
𝑠

𝜌 × 𝑉
𝑠

𝜌 × 𝐴 × 𝑙
𝑠
 𝜌 × 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐴𝐵𝐶𝐷 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑛 𝑥 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
= 𝜌𝐴𝑉  𝜌 × 𝑢 × (𝑑𝑦 × 𝑑𝑧)

6. 6
Mass of fluid leaving the face EFGH (outflow) per second,
CONTINUITY EQUATION IN 3
DIMENSIONS:
𝜌𝑢 𝑑𝑦𝑑𝑧 +
𝜕
𝜕𝑥
𝜌𝑢 𝑑𝑦𝑑𝑧 𝑑𝑥
Rate of increase in mass x-direction = mass through ABCD – mass through EFGH
= 𝜌𝑢 𝑑𝑦𝑑𝑧 − 𝜌𝑢 𝑑𝑦𝑑𝑧 −
𝜕
𝜕𝑥
𝜌𝑢 𝑑𝑦𝑑𝑧 𝑑𝑥
= −
𝜕
𝜕𝑥
𝜌𝑢 𝑑𝑦𝑑𝑧𝑑𝑥
= −
𝜕
𝜕𝑥
𝜌𝑢 𝑑𝑥𝑑𝑦𝑑z
Transport Phenomena

7. 7
Similarly,
Rate of increase in mass in y-direction = −
𝜕
𝜕𝑦
𝜌𝑣 𝑑𝑥𝑑𝑦𝑑z
Rate of increase in mass in y-direction = −
𝜕
𝜕𝑧
𝜌𝑤 𝑑𝑥𝑑𝑦𝑑z
➢ Total rate of increase in mass = −
𝜕
𝜕𝑥
𝜌𝑢 +
𝜕
𝜕𝑦
𝜌𝑣 +
𝜕
𝜕𝑧
𝜌𝑤 𝑑𝑥𝑑𝑦𝑑𝑧
CONTINUITY EQUATION IN 3
DIMENSIONS:
By the law of conversation of mass, there is no accumulation of mass
i.e mass is neither be created nor destroyed in the fluid element.
So net increase of mass per unit time in the fluid element must be
equal to the rate of increase of mass of fluid in the element.
...(1)
Transport Phenomena

8. 8
CONTINUITY EQUATION IN 3
DIMENSIONS:
Mass of fluid in the element is = 𝜌 𝑑𝑥𝑑𝑦𝑑𝑧 𝜌 =
𝑚
𝑉
𝑚 = 𝜌𝑉
its rate of increase with time
=
𝜕
𝜕𝑡
𝜌 𝑑𝑥𝑑𝑦𝑑𝑧
=
𝜕𝜌
𝜕𝑡
𝑑𝑥𝑑𝑦𝑑𝑧
Equating eq. (i) and (ii)
−
𝜕
𝜕𝑥
𝜌𝑢 +
𝜕
𝜕𝑦
𝜌𝑣 +
𝜕
𝜕𝑧
𝜌𝑤 𝑑𝑥𝑑𝑦𝑑𝑧 =
𝜕𝜌
𝜕𝑡
𝑑𝑥𝑑𝑦𝑑𝑧
…(2)
Transport Phenomena

9. 9
CONTINUITY EQUATION IN 3
DIMENSIONS:
𝜕𝜌
𝜕𝑡
= −
𝜕
𝜕𝑥
𝜌𝑢 +
𝜕
𝜕𝑦
𝜌𝑣 +
𝜕
𝜕𝑧
𝜌𝑤
This is the equation of continuity, which describes the time rate of
change of the fluid density at a fixed point in space.
This equation can be written more concisely by using vector
notation as follows
)
.
( V
t




−
=


Rate of increase
of mass per unit
volume
Net rate of addition of
mass per unit volume
by convection
Transport Phenomena

10. 
𝜕𝜌
𝜕𝑡
+
𝜕
𝜕𝑥
𝜌𝑢 +
𝜕
𝜕𝑦
𝜌𝑣 +
𝜕
𝜕𝑧
𝜌𝑤 = 0
▪ For steady state flow
𝜕𝜌
𝜕𝑡
= 0 and hence above equation becomes
as,
𝜕
𝜕𝑥
𝜌𝑢 +
𝜕
𝜕𝑦
𝜌𝑣 +
𝜕
𝜕𝑧
𝜌𝑤 = 0
▪ If the fluid is incompressible, then 𝜌 is constant and the above eq.
becomes as,
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
+
𝜕𝑤
𝜕𝑧
= 0
CONTINUITY EQUATION IN 3
DIMENSIONS:
Continuity equation in three-dimensions.
Transport Phenomena

11. Thanks!
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