As there's no step-by-step calculation or a single equation to calculate the volume and weight of a torispherical head in the ASME BPVC code, we had to find how to do it ourselves. Luckily we found an article in the Wolfram MathWorld[ 1 ] that helped us a lot. Below we'll explain how we used it to find out the inside volume and weight of the torispherical head used in the Shell & Head Design module.

The reason we don't put the equation in the Results page is because, as you'll see, it gets a little complicated and such calculation isn't required by ASME (to be honest, it's so easy to get the volume and weight using any CAD software that we believe many Users will just ignore the calculated value)

Summary

According to Wolfram MathWorld:

A torispherical dome is the surface obtained from the intersection of a spherical
cap with a tangent torus, as illustrated above. The radius of the sphere R is called
the "crown radius," and the radius a of the torus is called the "knuckle radius."
Torispherical domes are used to construct pressure vessels.

Fig. 2.1 - Torispherical Head Dimensions

Again, according to Wolfram MathWorld, the equation to find the volume (V) of a torispherical head is:

$$\begin{align}V_{head} = {{\pi \over 3}\Big[2hR^2-(2a^2+c^2+2aR)(R-h)+3a^2csin^{-1}{(R-h)\over(R-a)}\Big]}\tag{1}\end{align}$$To find the head height (h):

$$\begin{align}h = {{R - \sqrt{(a + c - R)(a - c - R)}}}\tag{2}\end{align}$$Where:

- R crown radius (it's the L variable in our application);
- h head height (calculated);
- a knuckle radius (it's the r variable in our application);
- c distance from the center of the head to the center of knuckle radius (D / 2 - r);

The Equation (1) don't take in account the straight flange of the head (V3 in Fig. 1.2):

$$\begin{align}V_{flange} = {{\pi \cdot D^2 \over 4} \cdot b}\tag{3}\end{align}$$Where:

- D inside diameter
- b straight flange height

Now we can calculate the __total inside volume__ of the head:

Fig. 1.2 - Torispherical Head Volumes

To find the weight we must subtract the total __inner__ volume (V1 + V3)
from the total __outer__ volume (V2 + V4). The outer volume must be calculated
using equation (1) and (3) and adding plate thickness (t) and corrosion allowance (ca)
to variables L, a R and D.

After finding the outer and inner volume we can now calculate the weight (Wg) of the torispherical head:

$$\begin{align}Wg = {{(V_{outer} - V_{inner}) * density}}\tag{5}\end{align}$$The material density can be found in ASME BPVC Section II[ 2 ] tables.

To check if our calculation is correct, we'll compare it with a CAD (Computer Aid Design) software, in this case, AutoCAD 2016[ 3 ].

Consider the following:

- R = 1000 mm
- a = 150 mm
- c = 350 mm
- t = 40 mm
- b = 60 mm
- density = 8000 kg/m^3

If c = 350 and a = 150, then D
= 1000 mm.

Results using equations (1), (2) and (3):

- $V_{inner}$ = 168354363.656 mm^3
- $V_{outer}$ = 223187593.836 mm^3
- Wg = 438.67 kg

Fig. 4.1 - AutoCAD 2016 Output for Inner Volume

Fig. 4.2 - AutoCAD 2016 Output for Outer Volume

As you can se in Figure 4.1 and 4.2, we got exactly the same result as CAD.

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