A vessel that operates at 800°F is supported by an insulated skirt. The thermal distribution in the skirt is shown in Fig. 5.24a. If the top and bottom of the skirt are assumed fixed with respect to rotation, what is the maximum stress due to temperature gradient? α = 7 × 10^-6 in./in.°F , μ = 0.3,

Question

A vessel that operates at 800°F is supported by an insulated skirt. The thermal distribution in the skirt is shown in Fig. 5.24a. If the top and bottom of the skirt are assumed fixed with respect to rotation, what is the maximum stress due to temperature gradient? α = 7 × 10[latex]^{-6}[/latex] in./in.°F , μ = 0.3, E = 30 × 10[latex]^6[/latex] psi.

Accepted Answer

The equation for linear temperature gradient is

[latex]T_{y}=T_{t}+\frac{T_{b}-T_{t}}{l} x[/latex]

The temperature change can be expressed as

[latex]T=\frac{T_{b}-T_{t}}{l} x[/latex]

and the circumferential stress due to ring action obtained from Eq. 5.30 a is

[latex]\sigma_{ heta}=\frac{-P r}{t}=-E \alpha T_{x}[/latex] (5.30a)

[latex]\frac{d^{4} w}{d x^{4}}+4 \beta^{4} w=\frac{E t \alpha T_{x}}{r D}[/latex] (5.30b)

[latex]\sigma_{ heta}=-E \alpha\left(\frac{T_{b}-T_{t}}{l} ight) x[/latex] (1)

Equation 5.30 b gives

[latex]\frac{d^{4} w}{d x^{4}}+4 \beta^{4} w=\frac{E t \alpha}{r D}\left(\frac{T_{b}-T_{t}}{l} ight) x[/latex]

A particular solution takes the form

[latex]w=c_{1} \frac{E t \alpha}{r D}\left(\frac{T_{b}-T_{t}}{l} ight) x+c_{2}[/latex]

which upon substituting into the differential equation gives

[latex]c_{1}=\frac{1}{4 \beta^{4}}[/latex] and [latex]c_{2}=0[/latex]

and w reduces to

[latex]w=r \alpha\left(\frac{T_{b}-T_{t}}{l} ight) x[/latex]

From Eq. 5.19

[latex]N_{ heta}=\frac{-E t w}{r}[/latex] (5.19)

[latex]\sigma_{ heta}=\frac{N_{ heta}}{t}=E \alpha\left(\frac{T_{b}-T_{t}}{l} ight) x[/latex] (2)

Adding Eqs. 1 and 2 results in

[latex]\sigma_{ heta}=0[/latex]

which means that for a linear distribution the thermal stress along the skirt is zero.

The slope due to axial gradient is given by

[latex] heta=\frac{d w}{d x}=r \alpha\left(\frac{T_{b}-T_{t}}{l} ight)[/latex]

Because the ends are fixed against rotation, a moment must be applied at the ends to reduce θ to zero. From Eq. 5.24

[latex]\begin{aligned}w &=\frac{1}{2 \beta^{3} D}\left(\beta M_{0} B_{\beta x}+Q_{0} C_{\beta x} ight) \\frac{d w}{d x} &=\frac{-1}{2 \beta^{2} d}\left(2 \beta M_{0} C_{\beta x}+Q_{0} A_{\beta x} ight) \\frac{d^{2} w}{d x^{2}} &=\frac{1}{2 \beta D}\left(2 \beta M_{0} A_{\beta x}+2 Q_{0} D_{\beta x} ight) \\frac{d^{3} w}{d x^{3}} &=\frac{-1}{D}\left(2 \beta M_{0} D_{\beta x}-Q_{0} B_{\beta x} ight)\end{aligned}[/latex] (5.24)

[latex]r \alpha\left(\frac{T_{b}-T_{t}}{l} ight)=\frac{-M_{0}}{\beta D}[/latex]

or

[latex]M_{0}=-\beta D\left(\frac{\alpha r}{l} ight)\left(T_{b}-T_{t} ight)[/latex]

Since [latex]\beta=0.2142, D=2.747 imes 10^{6}[/latex],

[latex]M_{0}=742 ext { in.- lb / in . }[/latex]

and

σ = 4450 psi

Question 5.14: A vessel that operates at 800°F is supported by an insulated...

Related Answered Questions

A pressure vessel operating at 300°F is subjected to a short excursion temperature of 600°F. At a given time, the temperature distribution in the wall is shown in Fig. 5.27. Find the maximum thermal stress at that instance. Let μ = 0.3, E = 30 × 10^6 psi, and α = 6.0 × 10^-6 in./in.°F ...

Verified Answer:

Verified Answer:

A cylindrical shell with r = 30 in. is simply supported at the ends. If L = 10 ft and t = 0.375 in., find the critical buckling pressure for a uniform applied pressure to sides and ends. Let E = 29 × 10^6 psi. ...

Verified Answer:

A pipe at 10°F is partly filled with liquid at 40°F and gas at 250°F as shown in Fig. 5.22 a. What is the maximum thermal stress if α = 6.5 × 10^-6 in./in.°F , E = 30 × 10^6 psi, and μ = 0.3 ? ...

Verified Answer:

Verified Answer:

Determine the maximum stress at point A of the thin cylinder in Fig. 5.14a. Let μ = 0.3. ...

Verified Answer:

Derive Nθ for the case of applied bending moment M0 at edge x = 0 for a short cylinder of length l. ...

Verified Answer:

A long cylindrical shell is subjected to end moment M0. Plot the value of Mx from βx = 0 to βx = 4.0. Also, determine the distance x at which the moment is about 7% of the original applied moment M0. ...

Verified Answer:

A cylinder has an inside radius of 72.0 in. and an internal pressure of 50 psi. What is the required thickness if the allowable stress is 15,000 psi, μ = 0.3, and E = 30 × 10^6 psi ? ...

Verified Answer:

The inside radius of a hydraulic cylinder is 12.0 in. What is the required thickness if P = 7500 psi and σθ = 20,000 psi? ...

Verified Answer: