Lecture Lecture Lecture Lecture 3_2024 new.pdf

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About This Presentation

Lecture Lecture Lecture


Slide Content

3D Modeling in Fluid Flow and Heat Transfer
MPE371
Dr. Mohammed Rabie
2023-2024
Mechanical Power Engineering, Mansoura University
1

2
Time required to empty a cylindrical tank
�=2 � ℎ

�
??????�
��
??????�
=ሶ�
�� −ሶ�
���
�
�
��= � (���)= �(
�
4
�
2
ℎ)
ሶ�
��=0
ሶ�
���= � ?????? �= �
�
4
�
2
2 � ℎ
??????�
�
4
�
2

??????�
=−�
�
4
�
2
2 � ℎ
�
2
−2 � �
2
�ℎ
��
= ℎ
�
2
−2 � �
2


0

−0.5
�ℎ= න
0
�
��
2 �
2
2 � �
2

0.5
=�

3
Time required to the level of fluid in a tank
??????= ?
�=2 � ℎ
ℎ2
�
�
ℎ1
�=2 � ℎ

Momentum equation in three dimensions
From Newton’s second law
�
��
��
= � ??????� ??????� ??????�
��
��
�
��
��
= � ??????� ??????� ??????�
��
��
�
��
��
= � ??????� ??????� ??????�
��
��
= ෍�
�
෍�
�
෍�
�
=
=
X-direction
Y-direction
Z-direction
=
??????� �
??????�
+∇.� � �??????� ??????� ??????�
??????� �
??????�
+∇.� � �??????� ??????� ??????�=
=
??????� �
??????�
+∇.� � �??????� ??????� ??????�

5
Forces on a Fluid Particle (Control Volume)
Surface Forces : Forces results from the interaction between
the control volume and the surroundings.
1. Pressure Force
2. Viscous Force
Body Forces : Forces associated with the volume of the control
volume.
1. Gravity Force (Weight)
2. Electromagnetic force (when the control volume is
subjected to electric or magnetic fields)

6
Body Forces
Gravity Force
�
�,??????= � റ�=� �
� �+ �
� �+�
� �
�
�,??????= � � �
� �+ �
� �+�
� �=� ??????� ??????� ??????� �
� �+ �
� �+�
� �
�
�,�=� �
� ??????� ??????� ??????�
�
�,�=� �
� ??????� ??????� ??????�
�
�,�=� �
� ??????� ??????� ??????�

7
Surface Forces
Pressure Forces
� �+
??????�
??????�
??????�
�
�+
??????�
??????�
??????�
�
�+
??????�
??????�
??????�
�=
�
??????
�
�,�=� −�+
??????�
??????�
??????�??????�??????�=−
??????�
??????�
??????�??????�??????�
�
�,�=� −�+
??????�
??????�
??????�??????�??????�=−
??????�
??????�
??????�??????�??????�
�
�,�=� −�+
??????�
??????�
??????�??????�??????�=−
??????�
??????�
??????�??????�??????�

8
Surface Forces
Viscous Forces
??????
��
??????
��
??????
��
??????
��
??????
��
??????
��
??????
��
??????
��
??????
��
??????=
??????
��??????
��??????
��
??????
��??????
��??????
��
??????
��??????
��??????
��
??????
�� ��������� �ℎ�� �ℎ� ������ ��������� ���� �� �ℎ� � ��??????��??????��� �� � ������� ������ �� �ℎ� � ��??????��??????���
??????
��
??????
��
??????
��
??????
��
??????
��
??????
��
??????
��
??????
��
??????
��

9
Fluid stresses (??????
��)
Deformation rate of the control volume (linear deformation rate and volumetric deformation rate)
??????
��= �
??????�
�
??????�
+
??????�
�
??????�
+ � ∇.�
??????
��=2�
??????�
??????�
+ � ∇.�
??????
��=2�
??????�
??????�
+ � ∇.�
??????
��=2�
??????�
??????�
+ � ∇.�
??????
��=??????
��= �
??????�
??????�
+
??????�
??????�
??????
��=??????
��= �
??????�
??????�
+
??????�
??????�
??????
��=??????
��= �
??????�
??????�
+
??????�
??????�

10
Momentum equation in three directions
??????� �
??????�
+∇.� � �=� �
�−
??????�
??????�
+∇.�∇�+[�
��]
??????� �
??????�
+∇.� � �=� �
�−
??????�
??????�
++∇.�∇�+[�
��]
??????� �
??????�
+∇.� w �=� �
�−
??????�
??????�
++∇.�∇�+[�
��]
Unsteady term
Convection term
Body Force
Pressure Force
Diffusion term
Viscous source
term
Viscous Force

@Osama Abdelrehim - Mohamed Sameh 11
Momentum equation in three directions

12
Flow between two horizontal parallel plates

13
??????�
??????�
+
??????��
??????�
+
??????��
??????�
+
??????��
??????�
=0
??????��
??????�
=0
??????��
??????�
+∇.���= −
??????�
??????�
+
????????????
��
??????�
+
????????????
��
??????�
+
????????????
��
??????�
+��
�
??????��
??????�
+u
??????��
??????�
+v
??????��
??????�
+�
??????��
??????�
= −
??????�
??????�
+
??????
??????�
�
??????�
??????�
+
??????
??????�
�
??????�
??????�
+
??????
??????�
�
??????�
??????�
+��
�

??????�
??????�
+
??????
??????�
�
??????�
??????�
=0
??????
2
�
??????�
2
=
1
�
??????�
??????�
??????�
??????�
=
1
�
??????�
??????�
�+�
1
�=
1
2 �
??????�
??????�
�
2
+�
1�+�
2

14
B.C.: �� �=0,�=0
�� �=ℎ,�=0
�
2=0 ����
1=
−1
2 �
??????�
??????�

�=
1
2 �
??????�
??????�
�
2
−1
2 �
??????�
??????�
ℎ �
�=
1
2 �
??????�
??????�
(�
2
− ℎ �)
Flow rate per unit width,
??????=න
0

� �??????=න
0

� � ��=න
0

� ��
??????=
−1
12 �
??????�
??????�

3
Mean velocity,
�
�??????��=
??????

=
−1
12 �
??????�
??????�

2
Maximum velocity,
??????�
??????�
=0
�=

2
�
���=
−1
8 �
??????�
??????�

2
=1.5�
�??????��
Flow rate per unit width b = 1 m

15
Energy equation
From the first law of thermodynamics,
�
d�
d�
= ∆�+∆�
Rate of work doneon a fluid particle
is equal to the product of the force
and the velocity component in the
direction of the forceNet rate of heat transfer to
the fluid particle
Energy of the fluid is the sum of
1.Internal energy
2.Kinetic energy
3.Potential energy

16
Energy equation
Viscous dissipation , results from the work
added to the control volume due to the
viscous force
Source term
+??????
�

@Osama Abdelrehim - Mohamed Sameh 17

18
1D Heat Conduction
Assumptions:
•Steady heat transfer
•No heat generation
•1-D heat flow (T=T(x))
Required:
1.Apply the assumptions to write down the heat transfer equation
2.Integrate the equation apply boundary conditions to find a
formula of the temperature in x-direction if the temperature at
the beginning of the rod = T
1, and at the end = T
2

19
zero =�
??????
2
�
??????�
2
+
??????
2
�
??????�
2
+
??????
2
�
??????�
2
??????
2
�
??????�
2
=0
by integration:
????????????
??????�
=�
1
After applying second integration: �=�
1�+�
2
B.C.: �� �=0,�=�
1
�� �=??????,�=�
2
�
1=
??????2−??????1
�
& �
2=�
1
�=
??????2−??????1
�
�+�
1
+??????
�

20
1D Heat Conduction with heat generation
zero =�
??????
2
�
??????�
2
+
??????
2
�
??????�
2
+
??????
2
�
??????�
2
+ �
�
zero =
??????
2
�
??????�
2
+
�
�
�
??????�
??????�
=−
�
�
�
�+ �
1
�=−
�
�
2 �
�
2
+�
1�+�
2
B.C.: �� �=0,�=�
1
�� �=??????,�=�
2
�
1=
??????2−??????1
�
+
????????????
�
?????? & �
2=�
1
�=−
????????????
2 �
�
2
+
??????2−??????1
�
+
????????????
�
??????�+�
1
+??????
�

Thanks…
21@Osama Abdelrehim