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2.An annular chemical reactor consists of a packed bed of catalyst between two coaxialcylinders. The inner and outer cylinders have radii of r0and r1, respectively. It isreasonable to assume that there is no heat transfer through the surface of the innercylinder, which is at a constant temperature T0. The catalytic reaction releases heat at auniform volumetric rate Sthroughout the reactor, whose effective thermal

conductivity kmay be considered constant. eglect the temperature gradients in theaxial direction.

a! "erive a second#order differential e$uation to describe the radial temperaturedistribution in the annular reactor starting with a shell thermal energy balance.

b! %stablish the radial temperature distribution by solving the differential e$uation.

c! &hat viscous flow problem is analogous to this heat conduction problem'

d! "erive an expression for the volumetric average temperature in the reactor.

e! "evelop an expression for the temperature at the outer cylindrical wall of the reactor.&hat will be the outer wall temperature if both the inner and outer radii are tripled'

a! (tep. "ifferential e$uation from thermal energy balance

)rom a thermal energy balance over a thin cylindrical shell of thickness rin the annular

reactor, we get

*ate of +eat In # ut - eneration / Accumulation

At steady#state, the accumulation term will be ero. (o,

(1)

where Sis the rate of generation of heat by chemical reaction per unit volume and qristhe heat flux in the radial direction.

"ividing by r Land taking the limit as rtends to ero,

(2)


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(3)

(ince the effective thermal conductivity kof the reactor bed may be considered

constant, on substituting )ourier2s law 3 ! we get

(4)

b!

(tep. *adial temperature profile by solving differential e$uation

n integrating,

(5)

The integration constants are determined using the boundary conditions4

(6)

(7)

The first boundary condition suggests no heat transfer through the inner cylindrical wallof the annulus.

n substituting the integration constants, the temperature profile is

(8)


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c!

(tep. Analogous problem in fluid mechanics

Figure.5elocity profile in falling film on circular tube is analogous to temperature profile

in annular chemical reactor.

The velocity profile for the falling film on the outside of a circular tube 3see )igure! isgiven by4

(9)

(ubstituting aR/ r0and R/ r1,

(10)

The maximum velocity 3which occurs at r/ r0! is

(11)


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The difference between the above two e$uations yields

(12)

%$uations 36! and 31! are identical in form. Thus, the analogous viscous flow problemis the laminar flow of a falling film on the insideof a circular tube. The e$uivalent$uantities are

(13)

d!

(tep. %xpression for volumetric average temperature

The volumetric average temperature in the reactor may be defined as

(14)

n substituting the temperature profile in the above expression and integrating

7using 8, we get

(15)

e!

(tep. %xpression for outer wall temperature

The temperature at the outer cylindrical wall 3r/ r1! of the reactor is given by

(16)


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&hen both the inner and outer radii are ntimes their original values, the term in s$uarebrackets gets multiplied by nand the outer wall temperature is thus given by

(17)

)or the case when both the radii are tripled, n/ 9 in the above expression.

1. An electric wire with radius r0of 0.:0 mm is made of copper 7electricalconductivity / :.1 x 10;ohm#1m#1and thermal conductivity / 960 &


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Figure.Thermal resistance representation of insulation and air film.

@ased on the above thermal resistance representation, the heat flow is

(1)

where kis the thermal conductivity of the plastic insulation.

(tep. +eat flow due to current in wire

The flow of an electric current results in some electrical energy getting converted tothermal energy irreversibly. The heat generation by electrical dissipation per unit volumeis given by S/ I


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)or the maximum current, the temperature T0must be maximied.

(tep. (ubstitution of numerical values

n setting the temperature T0to ?9.0o> 3i.e., the maximum allowable temperature for

the insulation!, the maximum current that can flow through the wire may be calculatedas 19.0; amp.

The numerical values substituted in the e$uation are given below.The values below may be changed and the problem solution recalculated with the newvalues provided in consistent units.

Variable name Symbol Value Unit

electrical conductivity ke :100000 ohm-1m-1

maximum temerature T0 ?9.0000o!

am"ient temerature T2 96.0000o!

outer radiu# r1 0.001:00 m

$ire radiu# r0 0.000:00 m

la#tic thermal conductivity k 0.9:000 %&(m ')

heat tran#er coeicient h 6.:0000 %&(m2')

Calculated Variable Symbol Value Unit

aximum current 13.027 am

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