Determine if the cooking process provides sufficient sterilization for the product with a long shelf life $(18$D$150$F$)$ if D$121$C$=30$s

t$=$ time required to heat the center to $150$F $($in seconds$),$

Carrots: t$=0\mathrm{.}000878996$

Potatoes: t$=0\mathrm{.}002617599$

Onions: t$=0\mathrm{.}002109591$

Thermal conductivity

Potato$=0\mathrm{.}68$ W$/($m$$K$).$

Onion $=0\mathrm{.}54$ W$/($m$$K$)$

Carrot $=0\mathrm{.}9$ W$/($m$$K$)$

Specific heat capacity $=3\mathrm{.}7$ J$/($g$$K$).$

Density of the vegetables $0\mathrm{.}95$ g$/$cm$^3.$

Soup temperature $=200$F$,$ and it remains constant.

Use the simplified heat transfer formula for conduction through a homogeneous material:

where:

t$=$rho$*$c$*$d$^2/(4$k$($Tfood$-$Tsoup$))$

t$=$ time required to heat the center to $150$F $($in seconds$),$

$=$ density of the food,

c $=$ specific heat capacity,

d $=$ half the thickness or half the side of the cube $($in meters$),$

k $=$ thermal conductivity,

Tfood$=$ final desired temperature of the vegetable center $(150$F$),$

Tsoup$=$ soup temperature $(200$F$).$