The dimensionless relationship.
Answer to Problem 50P
The dimensionless relationship is
Explanation of Solution
Given information:
The shaft power is
Write the expression of function of shaft power.
Write the dimension of density in
Here, the dimension of mass is
Write the dimension of the dynamic viscosity.
Here, the dimension of time is
Write the dimension of diameter in
Write the expression of angular velocity.
Here, the angle is
Write the dimension of time in
Write the dimension of angle in
Substitute
Write the expression of power.
Here, the energy is
Write the expression of Energy.
Here, the mass is
Write the dimension of mass in
Write the dimension of acceleration in
Write the dimension of displacement in
Substitute
Substitute
Write the dimension of tank diameter in
Write the dimension of average liquid depth in
Here, the number of variable is
Write the expression of number of pi-terms.
Here, the number of variable is
Substitute
Here, four pi-terms is present.
Here, the basic variable is
Write the expression of first pi-terms.
Here, the constants are
Write the expression of second pi-terms.
Write the expression of third pi-terms.
Write the expression of fourth pi-terms.
Write the dimension of first pi-term.
Write the dimension of second pi-term.
Write the dimension of third pi-term.
Write the dimension of fourth pi-term.
Substitute
Compare the power of
Compare the power of
Compare the power of
Substitute
Substitute
Compare the power of
Compare the power of
Compare the power of
Substitute
Substitute
Compare the power of
Compare the power of
Compare the power of
Substitute
Substitute
Compare the power of
Compare the power of
Compare the power of
Substitute
According to Buckingham pi-theorem the first pi-term is the function of rest of all another pi-term.
Substitute
Conclusion:
The dimensionless relationship is
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Chapter 7 Solutions
Fluid Mechanics: Fundamentals and Applications
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