B40DM FORMULAE AND CHARTS
Momentum Transport:
Equation of Continuity:
𝜕𝜌
𝜕𝜌
𝜕𝜌
𝜕𝜌
𝜕𝑣𝑥 𝜕𝑣𝑦 𝜕𝑣𝑧
+ 𝑣𝑥
+ 𝑣𝑦
+ 𝑣𝑧
+𝜌(
+
+
)=0
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
Navier-Stokes Equation:
•
Rectangular coordinates (x, y, z):
x-component:
𝜌
𝐷𝑣𝑥
𝜕𝑃
𝜕 2 𝑣𝑥 𝜕 2 𝑣𝑥 𝜕 2 𝑣𝑥
= 𝜌𝑔𝑥 −
+𝜇( 2 +
+
)
𝐷𝑡
𝜕𝑥
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
𝜌
𝐷𝑣𝑦
𝜕 2 𝑣𝑦 𝜕 2 𝑣𝑦 𝜕 2 𝑣𝑦
𝜕𝑃
= 𝜌𝑔𝑦 −
+𝜇( 2 +
+
)
𝐷𝑡
𝜕𝑦
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
y-component:
z-component:
𝜌
𝐷𝑣𝑧
𝜕𝑃
𝜕 2 𝑣𝑧 𝜕 2 𝑣𝑧 𝜕 2 𝑣𝑧
= 𝜌𝑔𝑧 −
+𝜇( 2 +
+
)
𝐷𝑡
𝜕𝑧
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
Heat Transport:
Convective-Diffusion Heat Equation:
•
Rectangular coordinates (x, y, z):
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕2𝑇
𝜕2 𝑇
𝜕2 𝑇
𝜌𝐶𝑝 ( 𝜕𝑡 + vx 𝜕x + vy 𝜕y + vz 𝜕z ) = k [𝜕𝑥 2 + 𝜕𝑦 2 + 𝜕𝑧 2 ] + 𝜙 + 𝑞̇
•
Cylindrical coordinates (r, 𝛉, 𝐳):
𝜌𝐶𝑝 (
𝜕𝑇
𝜕𝑇 vθ 𝜕𝑇
𝜕𝑇
𝜕 2 𝑇 1 ∂T 1 𝜕 2 𝑇 𝜕 2 𝑇
+ vr
+
+ vz ) = k [ 2 +
+
+
] + 𝜙 + 𝑞̇
𝜕𝑡
𝜕r
r 𝜕θ
𝜕z
𝜕r
r 𝜕r r 2 𝜕θ2 𝜕𝑧 2
•
Spherical coordinates (r, 𝛉, ∅):
𝜌𝐶𝑝 (
𝜕𝑇
𝜕𝑇 vθ 𝜕𝑇
v∅ 𝜕𝑇
)
+ vr
+
+
𝜕𝑡
𝜕r
r 𝜕θ rsin𝜃 𝜕∅
1 𝜕 2 ∂𝑇
1
𝜕
∂𝑇
1
𝜕 2𝑇
)+ 2
(sinθ ) + 2 2 ( 2 )] + 𝜙 + 𝑞̇
= k [ 2 (r
𝑟 𝜕𝑟
𝜕𝑟
𝑟 sin𝜃 𝜕𝜃
𝜕𝜃
𝑟 sin 𝜃 𝜕∅
Others:
ℎ𝐴𝑡
𝑇 − 𝑇∞
−
𝜌𝑉𝑐𝑝
=𝑒
𝑇𝑜 − 𝑇∞
ℎ𝑉
𝐵𝑖 = 𝐴
𝑘
𝛼𝑡
𝛼𝑡
=
2
(𝑉/𝐴)
(𝑥1 )2
𝐹𝑜 =
𝑌=
𝑇∞ − 𝑇
𝑇∞ − 𝑇𝑜
𝑋=
𝛼𝑡
𝑥1 2
𝑛=
𝑥
𝑥1
𝑚=
𝑘
ℎ𝑥1
𝑇𝑠 − 𝑇(𝑥, 𝑡)
𝑥
= erf⁡(
)
𝑇𝑠 − 𝑇𝑜
2√𝛼𝑡
𝑣=
𝜇
𝜌
𝛼=
𝑘
𝜌𝑐𝑝
𝑃𝑟 =
𝑣 𝜇𝑐𝑝
=
𝛼
𝑘
𝑁𝑢 =
ℎ𝐿
𝑘
Mass Transport:
•
Rectangular coordinates (x, y, z):
𝜕2𝑐
𝜕𝐶𝐴
𝜕2𝑐
𝜕2 𝑐
= 𝐷𝐴𝐵 [ 𝜕𝑥 2𝐴 + 𝜕𝑦 2𝐴 + 𝜕𝑧 2𝐴]
𝜕𝑡
•
Cylindrical coordinates (r, 𝛉, 𝐳):
𝜕𝐶𝐴
𝜕 2 𝑐𝐴 1 ∂𝑐𝐴 1 𝜕 2 𝑐𝐴 𝜕 2 𝑐𝐴
= 𝐷𝐴𝐵 [ 2 +
+
+
]
𝜕𝑡
𝜕r
r 𝜕r r 2 𝜕θ2
𝜕𝑧 2
•
Spherical coordinates (r, 𝛉, ∅):
𝜕𝐶𝐴
1 𝜕
∂𝑐𝐴
1
𝜕
∂𝑐𝐴
1
𝜕 2 𝑐𝐴
)+ 2
(sinθ
) + 2 2 ( 2 )]
= 𝐷𝐴𝐵 [ 2 (r 2
𝜕𝑡
𝑟 𝜕𝑟
𝜕𝑟
𝑟 sin𝜃 𝜕𝜃
𝜕𝜃
𝑟 sin 𝜃 𝜕∅
Others:
𝑁𝐴,𝑧 = −𝐷𝐴𝐵
𝑑𝑐𝐴 𝑐𝐴
+ (𝑁𝐴,𝑧 + 𝑁𝐵,𝑧 )
𝑑𝑧
𝑐
𝑌=
𝐶𝐴,∞ − 𝐶𝐴
𝐶𝐴,∞ − 𝐶𝐴,0
𝑋=
𝐷𝐴𝐵 𝑡
𝑥1 2
𝑛=
𝑥
𝑥1
𝑚=
𝐷𝐴𝐵
𝑘𝑐 𝑥1
𝐶𝐴,𝑠 − 𝐶𝐴 (𝑥, 𝑡)
𝑥
= erf⁡(
)
𝐶𝐴,𝑠 − 𝐶𝐴,0
2√𝐷𝐴𝐵 𝑡
𝑆𝑐 =
𝜈
𝜇
=
𝐷𝐴𝐵 𝜌𝐷𝐴𝐵
𝐿𝑒 =
𝛼
𝑘
=
𝐷𝐴𝐵 𝜌𝑐𝑝 𝐷𝐴𝐵
𝑆ℎ =
𝑘𝑐 𝐿
𝐷𝐴𝐵
𝑁𝐴 = 𝑘𝑐 (𝑐𝐴,𝑠 − 𝑐𝐴,∞ )
𝑘𝑐
𝑣∞
𝑘𝑐
𝑣∞
𝐶
ℎ
= 2𝑓 = 𝜌𝑣 𝑐
Sc=Pr=1
∞ 𝑝
=
𝐶𝑓 /2
𝑣𝑥
1+( )(𝑆𝑐−1)
𝑣∞
Pr=1
𝐶𝑓 /2
𝑆ℎ
𝑘𝑐
=
=
𝑅𝑒𝑆𝑐 𝑣∞ 1 + 5 × 𝐶 /2 {𝑆𝑐 − 1 + ln [1 + 5 (𝑆𝑐 − 1)]}
√ 𝑓
6
𝑘
𝐶
𝑗𝐷 = 𝑣 𝑐 𝑆𝑐 2/3 = 2𝑓 ⁡ 0.6<Sc<2500
∞
1
𝑆ℎ𝑥 = 0.332𝑅𝑒𝑥2 𝑆𝑐 2/3
Figure A.1 Central temperature history of various solids with initial uniform
temperature, To and constant surface temperature, Ts.
𝑥
Table A.1 Error Function erf (2 𝛼𝑡).
√
Figure A.2 Unsteady state transport in large flat slab.
Figure A.3 Unsteady state transport in a long cylinder.
Figure A.4 Unsteady state transport in a sphere.
Figure A.5 Center temperature history for an infinite plate.
Figure A.6 Center temperature history for an infinite cylinder.
Figure A.7 Center temperature history for a sphere.
Figure A.8 Charts for solution of unsteady transport problem: flat plate.
Figure A.9 Charts for solution of unsteady transport problem: cylinder.
Figure A.9 Charts for solution of unsteady transport problem: sphere.