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A certain mixture of reacting chemicals contained in a spherical tank generates heat at a rate of 168,746.9 W m^−3. If the inner and outer diameters of the tank wall are 99.5 cm and 100.2 cm, respectively, estimate the temperature drop (in °C) across the wall. The tank walls are made of steel with a conductivity of 18 W m^−1K^−1.

Answer :

The estimated temperature drop across the wall of the spherical tank is approximately 4.13 °C.

The temperature drop across the wall of the spherical tank can be estimated using the formula for heat conduction through a cylindrical wall. The formula is given by:

ΔT = (Q * r) / (4πkL)

where:

ΔT is the temperature drop in °C,

Q is the heat generation rate per unit volume (168746.9 W m^−3),

r is the radius of the tank wall (average of inner and outer radii) in meters,

k is the thermal conductivity of the steel (18 W m^−1K^−1),

L is the thickness of the tank wall in meters.

To calculate the radius of the tank wall (r):

r = (99.5 cm + 100.2 cm) / 2

= 99.85 cm = 0.9985 m

Assuming the thickness of the tank wall (L) is negligible compared to the radius, we can use this simplified formula:

ΔT = (Q * r) / (4πk)

Substituting the given values into the formula, we have:

ΔT = (168746.9 * 0.9985) / (4π * 18)

Calculating the result:

ΔT = 466.84 / (4π * 18)

≈ 4.13 °C

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