A cylindrical pressure vessel with 1 m inner diameter is subjected to internal steam pressure of 1.5 MPa. The permissible stresses for the cylinder plate and the rivets in tension, shear and compression are 80, 60 and 120 N/mm² respectively. The efficiency of longitudinal joint can be taken as 80% for the purpose of calculating the plate thickness. The efficiency of circumferential lap joint should be at least 62%. Design the circumferential lap joint and calculate:
(i) thickness of the plate;
(ii) diameter of the rivets;
(iii) number of rivets;
(iv) pitch of rivets;
(v) number of rows of rivets; and
(vi) overlap of the plates.
Question 8.23: A cylindrical pressure vessel with 1 m inner diameter is sub...
- Inner diameter of cylindrical pressure vessel: 1 m
- Internal steam pressure: 1.5 MPa
- Permissible stress for cylinder plate in tension: 80 N/mm²
- Permissible stress for cylinder plate in shear: 60 N/mm²
- Permissible stress for cylinder plate in compression: 120 N/mm²
- Efficiency of longitudinal joint: 80%
- Minimum efficiency of circumferential lap joint: 62%
Step 1:
Thickness of plate (Eq. 8.43) To calculate the thickness of the plate, we use the equation: t = (P_i D_i) / (2 σ_t * η) + CA where t is the thickness of the plate, P_i is the internal pressure, D_i is the diameter of the vessel, σ_t is the tensile stress, η is the efficiency, and CA is the corrosion allowance.
Step 2:
Diameter of rivets (Eq. 8.45) We calculate the diameter of the rivets using the equation: d = 6 * √t where d is the diameter of the rivets and t is the thickness of the plate.
Step 3:
Number of rivets (Eq. 8.58) The number of rivets is calculated using the equation: n = (D_i / d)^2 * (P_i / τ) where n is the number of rivets, D_i is the diameter of the vessel, d is the diameter of the rivets, P_i is the internal pressure, and τ is the shear stress.
Step 4:
Pitch of rivets (Eq. 8.59, 8.48, 8.49) We calculate the pitch of the rivets using the equation: η_1 = (p_1 - d) / p_1 where η_1 is the pitch efficiency and p_1 is the pitch of the rivets. The minimum pitch is given by p_min = 2d and the maximum pitch is given by p_max = C*t + 41.28.
Step 5:
Number of rivets in one row (Eq. 8.60) The number of rivets in one row is given by: n_1 = (π * (D_i + t)) / p_1
Step 6:
Number of rows of rivets Assuming a single-riveted lap joint, the number of rows of rivets is one.
Step 7:
Revised value of rivet diameter (Eq. 8.58) We recalculate the diameter of the rivets using the equation: n = (D_i / d)^2 * (P_i / τ)
Step 8:
Overlap of plates (Eq. 8.62) The margin, m, is given by: m = 1.5d The overlap of plates, a, is given by: a = p_t + 2m
Step 9:
Check for design (Eq. 8.59) We check the efficiency of the joint using the equation: η_1 = (p_1 - d) / p_1
By following these steps, we can determine the necessary parameters for the design of the vessel and ensure that it meets the required specifications.
Final Answer
\text { Given For vessel, } D_{i}=1 m \quad P_{i}=1.5 MPa .
\sigma_{t}=80 N / mm ^{2} \quad \tau=60 N / mm ^{2} \quad \sigma_{c}=120 N / mm ^{2} .
\text { For longitudinal joint, } \eta=80 \% .
\text { For circumferential joint, } \eta_{1}=62 \% .
Step I Thickness of plate
From Eq. (8.43),
t=\frac{P_{i} D_{i}}{2 \sigma_{t} \eta}+ CA (8.43).
t=\frac{P_{i} D_{i}}{2 \sigma_{t} \eta}+ CA =\frac{1.5(1000)}{2(80)(0.8)}+2 .
= 13.72 or 14 mm (i)
Step II Diameter of rivets
t > 8 mm
From Eq. (8.45),
d=6 \sqrt{t} (8.45).
d=6 \sqrt{t}=6 \sqrt{14}=22.45 \quad \text { or } \quad 23 mm.
Step III Number of rivets
From Eq. (8.58),
n=\left(\frac{D_{i}}{d}\right)^{2} \frac{P_{i}}{\tau} (8.58).
n=\left(\frac{D_{i}}{d}\right)^{2} \frac{P_{i}}{\tau}=\left(\frac{1000}{23}\right)^{2} \frac{(1.5)}{(60)}=47.26 \text { or } 48 .
Step IV Pitch of rivets
From Eq. (8.59),
\eta_{1}=\frac{p_{1}-d}{p_{1}} (8.59).
\eta_{1}=\frac{p_{1}-d}{p_{1}} \quad \text { or } \quad 0.62=\frac{p_{1}-23}{p_{1}} .
\therefore \quad p_{1}=60.53 \text { or } 62 mm (a).
From Eqs (8.48) and (8.49),
p_{\min .}=2 d (8.48).
p_{\max }=C t+41.28 (8.49).
p_{\min .}=2 d=2(23)=46 mm (b).
p_{\max .}=C t+41.28=1.31(14)+41.28 .
= 59.62 mm (c)
From (a) and (c),
p_{1}>p_{\max } .
The pitch of rivets should be from 46 mm to 59.62 mm. We will assume the pitch as 55 mm and recalculate the number of rivets and diameter of rivet.
p_{1}=55 mm (iv)
From Eq. (8.60), the number of rivets in one row is given by,
n_{1}=\frac{\pi\left(D_{i}+t\right)}{p_{1}} (8.60).
n_{1}=\frac{\pi\left(D_{i}+t\right)}{p_{1}}=\frac{\pi(1000+14)}{55}=57.92 \text { or } 58 (iii).
Step V Number of rows of rivets
It is assumed that the type of joint is single-riveted lap joint. The number of rows of rivets is one.
From Eq. (8.58), revised value of rivet diameter is obtained.
n=\left(\frac{D_{i}}{d}\right)^{2} \frac{P_{i}}{\tau} (8.58).
n=\left(\frac{D_{i}}{d}\right)^{2} \frac{P_{i}}{\tau} \quad \text { or } \quad 58=\left(\frac{1000}{d}\right)^{2} \frac{(1.5)}{(60)} .
d = 20.76 or 21 mm (ii)
Step VI Overlap of plates
The margin m is given by,
m = 1.5d = 1.5 (21) = 31.5 or 35 mm
From Eq. (8.62),
a=p_{t}+2 m (8.62).
a=p_{t}+2 m=0+2(35)=70 mm (vi).
Step VII Check for design
From Eq. (8.59),
\eta_{1}=\frac{p_{1}-d}{p_{1}} (8.59).
\eta_{1}=\frac{p_{1}-d}{p_{1}}=\frac{55-21}{55}=0.6182 \text { or } 61.82 \% .
The efficiency of the joint is very near to 62% and no changes are required in the calculations.
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