Download Reynolds Transport Theorem

ME 362
Reynolds Transport Theorem
Page 1 of 2
Reynolds Transport Theorem:
Type of problems involved:


Fixed or moving CV
Single or multiple inflow(s) and outflow(s)
Solution Procedure:
For a fixed CV case:
Step 1: Assumption


Assume steady and incompressible flow
Neglect gravity and viscous effect
Step 2: CV Analysis

Sketch CV and include coordinate system
Step 3: Mass Conservation

 m in   m out
where m  AV and V is the velocity entering/leaving the CV
Solve for Vout , m in and m out .
Step 4: Momentum Conservation




 V ) out  (m
 V ) in
 F  (m

where F is the reaction force from the cart

Solve for F  Fx , Fy  .
Step 5: Power Calculation

P   FxVc  0
where P is power delivered to the cart

dP
0
dVC
when P is maximum
ME 362
Reynolds Transport Theorem
Page 2 of 2
For a moving CV with constant velocity Vc case:


Follow the same steps as in the fixed CV case
Remember we are dealing with the relative velocity Vr now
The following table shows the differences in the governing equations between the fixed
CV case and the moving CV case.
Fixed CV
 m in   m out
where m  AV
Mass Conservation
Momentum Conservation



 V ) out  (m
 V ) in
 F  (m
Moving CV

with constant velocity VC
 r ) in  (m
 r ) out
(m
 r  AVr
where m
Vr  V  VC



 rVr ) out  (m
 rVr ) in
 F  (m
Example:
A water jet of velocity Vj strikes a vane mounted on a frictionless cart that moves at
constant velocity Vc as shown in Fig. 1. The water jet is split equally by the vane, where
Aj is the cross-sectional area of the jet. Neglect the gravity and viscous effect.

1) Compute the force F  Fx , Fy  required to keep the cart moving at constant velocity.
2) Find the velocity Vc for which the power P delivered to the cart is a maximum.
Aj

Vj
90
Aj
Aj
Vc
Fy
Fig. 1
Fx