Dimensional Analysis - Similitude

 

Example:   At a large fish hatchery the fish are reared in open, water-filled tanks.  Each tank is

Approximately square in shape with curved corners, and the walls are smooth.  To create motion in

the tanks, water is supplied through a pipe at eh edge of the tank.  The water is drained from the

tank through an opening at the center.  A model with a length scale of 1:13 is to be used to find the

fluid velocity, V,   at various locations  within the tank.

Assume that V is a function of the following variables:  L, L1, ρ, μ, g, and Q
where  L  is some characteristic length
            L1 represents a series of other pertinent lengths such as the tank width such as inlet
            pipe diameter, etc
            ρ is the fluid density 
            μ is the fluid viscosity
            g is the acceleration of gravity,
           Q is the discharge through the tank

i.e.    V  =  f(L, L1, ρ, μ, g, Q)

a. Determine a suitable set of dimensionless products for this application and the prediction
equation for the velocity, V.  If water is to be used for the model, can all of the similarity
requirements be met? 

b. If the flowrate, Q, into the full-sized tank is 250 gpm, find the required value for the model
discharge assuming Froude number similarity.  What model depth will correspond to a depth of
32 in. in the full-sized tank?       

Strategy:      Need to find dimensionless products first.                                          

1.

To start identify variables governing the problem.  (Given in problem statement.)

 

 

2.

List dimensions for each variable.  Pick  F,L, and T  as the set of dimensions to

form dimensionless products.  Below are the dimensions for each variable.

 

V → L/T,    L → L,     L1→ L,     ρ → FT2/L4,     μ→ FT/L2,    g  → L/T2,    Q→L3/T

 

Note:  μ comes from  τ = μ dv/dy

3.

Number of dimensionless products (Pi terms) =  4     since

       number of variables =  7    number of dimensions =  3

4.

Select the dependent variable and the repeating variables.

 

Dependent variable = V,    Non-repeating variables:  L1, μ, and Q

 

Select the following repeating variables:  L for L,  ρ for F,  g for T

 

Click here to continue with this example.

 


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