Question

Question #97470

Answer 1

It is clear that the slab of steel at `70^@C` will loose heat via all three modes: Conduction, Convection and Radiation simultaneously and finally will be at ambient temperature given as `10^@C`.

`"Total heat lost by the steel slab"=mst`
where `m` is mass of slab, `s` specific heat of steel and `t` is change in its temperature.

Assuming density `rho` of steel`=7850kgm^-3`, `s=511Jkg^-1K^-1`
`m=rhoxx"volume"`
`m=7850xx(0.01xx1xx3)=235.5kg`

`:."Total heat lost by the steel slab"=235.5xx511xx(70-10)=7220430J`
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A. Now, rate of heat loss due to conduction `dotQ` from slab to deck is calculated as below.

Lets assume that the `1.0 cm` thick slab is lying flat on the deck.

Deck is at temperature of surroundings`=10^@C`

The heat loss from top to bottom of slab is found by taking
`ΔT = 70^@C – 10^@C = 60^@C = 60 K`, the temperature difference between the slab and deck.

Area `A` through which heat is lost`=1xx3=3m^2`

Thermal conductivity `k("steel") = 50Wm^-1K^-1`

We know that basic conduction problem equation is

`dot Q=(kAΔT)/"thickness of slab"`

Inserting above values we get
`dot Q=(50xx3xx60)/0.01`
`=>dot Q=900000W`

B. Similarly basic convection problem equation is
basic equation for convection,

`dotQ=hAΔT`
where heat transfer coefficient `h=10Wm^-2K^-1`.

Area `A` in contact with air`=2xx0.01(3+1)+3xx1=3.08m^2`

Inserting above values we get
`dotQ=10xx3.08xx60=1848W`

C. For radiation loss we have Stefan-Boltzmann Law.

`dotQ=εAσ(T_1^4– T_2^4)`
where `sigma` is the Stefan Boltzmann constant `= 5.670367 xx 10^-8 Wm^-2 K^-4`, `T_1K` is temperature of radiating body and `T_2K` is temperature of surroundings.

We are given the emissivity for steel, `ε= 0.85`. The radiating surface area as calculated above, `A= 3.08 m^2`
Inserting given values we get
`dotQ=0.85xx3.08xx5.670367 xx 10^-8(343^4– 283^4)=1102W`

D. Total heat loss rate `=900000+1848+1102=902950W`

Assuming rate of heat loss via each mode is constant throughout the time of loss.
Heat lost via conduction`=7220430xx900000/902950approx7196840J`