Problem: Convection$-$Conduction in a Combined Multi$-$Layer Insulated System

Problem Statement:

A chemical processing plant utilizes a combination of a multi$-$layer insulated wall truck and a

multi$-$layer insulated pipe to maintain the temperature of a fluid during its transport and

storage to another facility.

System Description:

$1.$ Storage Tank Wall:

o The storage tank has a multi$-$layer wall consisting of:

$$ Layer $1($inner layer$)$: A $0\mathrm{.}04$ m thick layer of stainless steel with

thermal conductivity k$1=16$W$/$m$$K$.$

$$ Layer $2($middle layer$)$: A $0\mathrm{.}06$ m thick layer of fiberglass insulation

with thermal conductivity k$2=0\mathrm{.}035$ W$/$m$$K$.$

$$ Layer $3($outer layer$)$: A $0\mathrm{.}03$ m thick layer of aluminum with thermal

conductivity k$3=205$W$/$m$$K$.$

o The inside of the tank is filled with a hot fluid at T$\backslash$infty $,1=150\backslash$deg C with a convective

heat transfer coefficient h$1=200$ W$/$m$^2$K$.$

o The outer surface of the tank is exposed to ambient air at T$\backslash$infty $,2=25\backslash$deg C with a

convective heat transfer coefficient h$2=15$W$/$m$^2$K$.$

o The total surface area of the tank wall is A wall$=40$ m$^2.$

$2.$ Insulated Transport Pipe:

o A cylindrical pipe with three concentric layers is used to transport the same

fluid from the storage tank to another processing unit. The layers of the pipe

are:

$$ Layer $1($innermost layer$)$: A $0\mathrm{.}008$ m thick layer of copper with

thermal conductivity k$4=385$ W$/$m$$K$.$

$$ Layer $2($middle layer$)$: A $0\mathrm{.}05$ m thick layer of mineral wool insulation

with thermal conductivity k$5=0\mathrm{.}042$ W$/$m$$K$.$

$$ Layer $3($outermost layer$)$: A $0\mathrm{.}015$ m thick layer of rubber with

thermal conductivity k$6=0\mathrm{.}15$ W$/$m$$K$.$

o The pipe has an inner radius r$1=0\mathrm{.}05$ and a length of L$=20$L $=20$L$=20$ m$.$

o The inside of the pipe is exposed to the same hot fluid at T$\backslash$infty $,3=150\backslash$deg C with a

convective heat transfer coefficient h$3=250$ W$/$m$^2$K$.$

o The outer surface of the pipe is exposed to ambient air at T$\backslash$infty $,4=25\backslash$deg C with a

convective heat transfer coefficient h$4=12$W$/$m$^2$K$.$Tasks:

$1.$ Calculate the thermal resistances for each layer $($conduction$)$ and for convection on

both the inner and outer surfaces of the storage tank wall.

$2.$ Calculate the thermal resistances for each cylindrical layer $($conduction$)$ and for

convection on both the inner and outer surfaces of the pipe.

$3.$ Determine the total thermal resistance for both the tank wall and the pipe.

$4.$ Calculate the steady$-$state heat transfer rates through the storage tank wall and the

pipe.

$5.$ Evaluate the effectiveness of the insulation: If the acceptable heat loss is $800$ W$,$

determine which system is better for the chemical processing plant.