Design of a high$-$temperature high$-$pressure processing vessel

$[150$ Points Total$]$

In order to control bacterial growth and prevent food$-$borne illness, many food products $($especially liquid products such as juices, salsa, etc.$)$ are treated before sale with a technique called high$-$pressure processing $($HPP$).$ As the name implies, this process consists of placing the food product in a pressure$-$transmitting medium $($usually water$)$ inside a gigantic pressure vessel and pressurizing it to $400-600$ MPa, which has a lethal effect on many $($but not all$)$ bacteria. Recently, the industry has started to merge conventional HPP with thermal processing $($think pasteurization$)$ in a technique called High$-$Pressure Thermal Processing $($HPTP$),$ which involves simultaneous application of ~$40130\backslash$deg C and $400-600$ MPa.

With the addition of this thermal component, the stress of analysis of the chamber needs to be re$-$evaluated, and a large food company has employed YOU to perform this evaluation. Let$$s get started!

$1.$ Pressure vessel design for room temperature High$-$Pressure Processing $[70$ points$]$

First, let$$s establish a quantitative description of conventional HPP pressure vessel design $($i$.$e$.$ without the thermal component$).$ Let$$s assume that your pressure vessel is comprised of a very long $($i$.$e$.$ length $>>$ outer diameter$),$ thick$-$walled cylinder, which is pressurized by two large pistons at either end $($Figure $1).$ In this case, we design pressure vessels based on the assumption that they will fail due to the total stress from radial and tangential $($aka $$Hoop$)$ stress contributions. Note that when the wall thickness of a cylinder pressure vessel is about one$-$tenth, or less, of its inner radius, it is considered a thick$-$wall pressure vessel.

Your goal is to design the least expensive vessel possible, that will tolerate a given pressure P$.$ However, your metals vendor has not yet given you cost$/$kg figures. As such, begin by assuming that all materials will cost the same amount per kg$.$

Using the GIVEN INFORMATION at the end of this document, in addition to any other expertise you have in stress analysis, do the following:

a$.$ Plot the non$-$dimensional stress $($sigma $/$P$)$ as a function of non$-$dimensional radius $($r$/$ro$)$ for the hoop stress, the radial stress, and the total stress, on the same plot. Note at which radial position the stress is highest.

GIVEN INFORMATION:

Tangential stress $($Hoop stress$)$ in a thick$-$wall pressure vessel as a function of $r$ :

${}_h=\frac{r_i^{2}P}{r_o^2-r_i^2}(1+\frac{r_o^2}{r^2})$

Radial stress in a thick$-$wall pressure vessel as a function of $r$ :

${}_r=\frac{r_i^{2}P}{r_o^2-r_i^2}(1-\frac{r_o^2}{r^2})$

Where $r_o$ is the outer radius and $r_i$ is the inner radius of the chamber, and $P$ is the pressure.

Total Stress:

${}_t=\sqrt[\phantom{2}]{{}_r^2+{}_h^2}$