In a direct acting pressure relief valve, 64 bar pressure is acting on the conical dart seating area and the dart is kept closed by a spring force of 50 kg opposing the pressure. (5)
a) Determine the bore diameter on the seat.
b) What will the spring force be if a pressure of 32 bar is acting on diameter of 10 mm seating area?
a) To determine the bore diameter on the seat, we can use the formula for the force exerted on the seat of a pressure relief valve:
F = P × A
Where F is the force, P is the pressure, and A is the area.
Given:
Pressure (P) = 64 bar
Force (F) = 50 kg (which can be converted to Newtons by multiplying by the acceleration due to gravity, 9.81 m/s²)
We need to determine the area (A) and subsequently the bore diameter.
Rearranging the formula, we have:
A = F / P
Substituting the values, we get:
A = (50 kg × 9.81 m/s²) / (64 bar)
Now, let's convert the pressure from bar to Newtons per square meter (N/m² or Pa) by multiplying by 10⁵:
A = (50 kg × 9.81 m/s²) / (64 × 10⁵ N/m²)
Simplifying the expression, we have:
A = 0.7641 m²
The area (A) of the seating is given by the formula:
A = πr²
Where r is the radius of the bore.
Rearranging the formula, we have:
r = √(A / π)
Substituting the value of A, we get:
r = √(0.7641 m² / π)
Calculating this, we find:
r ≈ 0.49 m
Finally, the bore diameter (d) is twice the radius (r), so:
d = 2 × r
Substituting the value of r, we get:
d ≈ 2 × 0.49 m
Calculating this, we find:
d ≈ 0.98 m
b) To find the spring force if a pressure of 32 bar is acting on a seating area of diameter 10 mm, we need to calculate the force exerted on the seat using the formula:
F = P × A
Given:
Pressure (P) = 32 bar
Area (A) = πr²
Diameter (d) = 10 mm
First, we need to convert the diameter to meters:
d = 10 mm = 0.01 m
Now, we can find the radius (r) by dividing the diameter by 2:
r = 0.01 m / 2
Calculating this, we find:
r = 0.005 m
Using the formula for the area of a circle:
A = π (0.005 m)²
Calculating this, we find:
A ≈ 0.0000785398 m²
Now, substituting the values into the formula for the force, we have:
F = (32 bar) × (0.0000785398 m²)
Converting the pressure from bar to Newtons per square meter (N/m² or Pa) by multiplying by 10⁵, we get:
F = (32 × 10⁵ N/m²) × (0.0000785398 m²)
Calculating this, we find:
F ≈ 251.327412 N
a) To determine the bore diameter on the seat, we need to use the equation for the spring force opposing the pressure.
The formula for the spring force opposing the pressure in a direct acting pressure relief valve is:
Spring Force = Pressure * Seat Area
Given that the pressure acting on the conical dart seating area is 64 bar and the spring force opposing the pressure is 50 kg, we can rearrange the formula to solve for the seat area.
Seat Area = Spring Force / Pressure
Seat Area = 50 kg / 64 bar
To convert the pressure from bar to Pascal, we need to multiply by 10^5 since 1 bar is equal to 10^5 Pascal.
Seat Area = 50 kg / (64 bar * 10^5 kg/m^2/bar)
Now we can calculate the seat area in m^2. Since the seat is conical, we need to consider the diameter of the seat.
Seat Area = π * (Seat Diameter^2) / 4
Let's solve for the seat diameter:
(π * (Seat Diameter^2) / 4) = 50 kg / (64 bar * 10^5 kg/m^2/bar)
Rearranging the formula to solve for the seat diameter:
Seat Diameter^2 = (4 * 50 kg / (π * 64 bar * 10^5 kg/m^2/bar)
Seat Diameter^2 = (4 * 50 kg * 10^-5 bar^-1) / (π * 64)
Seat Diameter^2 = (200 kg * 10^-5 bar^-1) / (π * 64)
Seat Diameter^2 ≈ 9.9 * 10^-6 m^2
Now we can calculate the seat diameter:
Seat Diameter ≈ √(9.9 * 10^-6 m^2)
Seat Diameter ≈ 0.0032 m
Therefore, the bore diameter on the seat is approximately 0.0032 m, or 3.2 mm.
b) To determine the spring force if a pressure of 32 bar is acting on a seat diameter of 10 mm, we can use the same formula for the spring force:
Spring Force = Pressure * Seat Area
First, we need to calculate the seat area:
Seat Area = π * (Seat Diameter^2) / 4
Seat Area = π * (0.01 m^2) / 4
Seat Area ≈ 7.85 * 10^-5 m^2
Now we can calculate the spring force:
Spring Force = 32 bar * (7.85 * 10^-5 m^2)
To convert the pressure from bar to Pascal:
Spring Force = 32 bar * (7.85 * 10^-5 m^2 * 10^5 kg/m^2/bar)
Spring Force = 32 bar * (7.85 * 10^0 kg)
Spring Force ≈ 250.72 kg
Therefore, the spring force if a pressure of 32 bar is acting on a seat diameter of 10 mm is approximately 250.72 kg.