Question 1.9: Show that the kinetic energy of a body equals kmV² using the......

Turbomachinery Design and Theory [1005770]

Show that the kinetic energy of a body equals kmV² using the method of dimensional analysis.

Step-by-Step

The 'Blue Check Mark' means that this solution was answered by an expert.
Learn more on how do we answer questions.

Since the kinetic energy of a body depends on its mass and velocity,
K.E. = f (V, m), or K.E. = $kV^a m^b$ .
Dimensionally,
$FLT^0 = (LT^{-1})^a(FT^2L^{-1})^b$
Equating the exponents of F, L, and T, we get:
F: 1 = b; L: 1 = a – b; T: 0 = – a + 2b
This gives b = 1 and a = 2. So, K.E. = kV²m, where k is a constant.

Related Answered Questions

Question: 1.1

A radial flow hydraulic turbine produces 32 kW under a head of 16mand running at 100 rpm. A geometrically similar model producing 42 kW and a head of 6mis to be tested under geometrically similar conditions. If model efficiency is assumed to be 92%, find the diameter ratio between the model and ...

Verified Answer:

Assuming constant fluid density, equating head, fl...

Question: 1.10

Consider a radial inward flow machine, the radial and tangential velocity components are 340 m/s and 50 m/s, respectively, and the inlet and the outlet radii are 14 cm and 7 cm, respectively. Find the torque per unit mass flow rate. ...

Verified Answer:

Here,

r_1

= 0.14 m

C_{w1}[/l...

Question: 1.8

At what velocity should tests be run in a wind tunnel on a model of an airplane wing of 160 mm chord in order that the Reynolds number should be the same as that of the prototype of 1000 mm chord moving at 40.5 m/s. Air is under atmospheric pressure in the wind tunnel. ...

Verified Answer:

Let Velocity of the model :

V_m

...

Question: 1.7

Consider an axial flow gas turbine in which air enters at the stagnation temperature of 1050 K. The turbine operates with a total pressure ratio of 4:1. The rotor turns at 15500 rpm and the overall diameter of the rotor is 30 cm. If the total-to-total efficiency is 0.85, find the power output per ...

Verified Answer:

Using the isentropic P – T relation:

T^′_{0...

Question: 1.6

Consider an axial flow pump, which has rotor diameter of 32 cm that discharges liquid water at the rate of 2.5 m³/min while running at 1450 rpm. The corresponding energy input is 120 J/kg, and the total efficiency is 78%. If a second geometrically similar pump with diameter of 22 cm operates at ...

Verified Answer:

Using the geometric and dynamic similarity equatio...

Question: 1.2

A centrifugal pump delivers 2.5 m³/s under a head of 14 m and running at a speed of 2010 rpm. The impeller diameter of the pump is 125 mm. If a 104 mm diameter impeller is fitted and the pump runs at a speed of 2210 rpm, what is the volume rate? Determine also the new pump head. ...

Verified Answer:

First of all, let us assume that dynamic similarit...

Question: 1.5

Prove that the drag force F on a partially submerged body is given by: where V is the velocity of the body, l is the linear dimension, r, the fluid density, k is the rms height of surface roughness, and g is the gravitational acceleration. ...

Verified Answer:

Let the functional relation be: F = f (V, l, k,ρ, ...

Question: 1.4

pump discharges liquid at the rate of Q against a head of H. If specific weight of the liquid is w, find the expression for the pumping power. ...

Verified Answer:

Let Power P be given by:

P = f(w, Q, H) = k...

Question: 1.3

An axial flow compressor handling air and designed to run at 5000 rpm at ambient temperature and pressure of 18°C and 1.013 bar, respectively. The performance characteristic of the compressor is obtained at the atmosphere temperature of 25°C. What is the correct speed at which the compressor must ...

Verified Answer:

Since the machine is the same in both cases, the g...