# Minimum Design Metal Temperature

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)

### 1. What's a Torispherical Head? top

According to Wolfram MathWorld:

### 2. Torispherical Head Inside Volume top

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:

\begin{align}V_{total} = {V_{head} + V_{flange}}\tag{4}\end{align}
Fig. 1.2 - Torispherical Head Volumes

### 3. Torispherical Head Weight top

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.

### 4. Calculation Checking top

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

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.

Reference top