a. The maximum thickness of the copper nozzle is 10.71 cm
b. The maximum thickness of the steel nozzle is 0.35 cm
The question has to do with heat transfer
<h3>What is heat transfer?</h3>
Heat transfer is the movement of heat energy from one body to anotrher.
<h3>How to calculate the maximum wall thickness?</h3>
Since the rate of heat loss by the gas equal rate of heat gain by the metal.
<h3>Rate of heat loss by gas</h3>
The rate of heat loss by gas is P = -hA(T - T') where
- h = heat transfer coefficient of gas = 2 × 10⁴ W/m²-K,
- A = surface area of nozzle,
- T = maximum temperature of metal and
- T = Temperature of gas = 2750°C
<h3>Rate of heat gain by metal</h3>
The rate of heat gain by metal is given by P' = kA(T - T")/t where
- k = thermal coefficient of metal,
- A = surface area of nozzle,
- T = maximum temperature of metal,
- T" = temperature of exterior wall of nozzle = 150°C and
- t = thickness of nozzle.
<h3>Maximum thickness of nozzle.</h3>
Since P = P', we have that
-hA(T - T') = kA(T - T")/t
Making t subject of the formula, we have
t = -k(T - T")/h(T - T')
<h3>a. Maximum thickness for copper nozzle</h3>
Given that for copper
- T = 540°C and
- k = 378 W/m-K
Substituting the values of the variables into t, we have
t = -k(T - T")/h(T - T')
t = -378 W/m-K(540°C - 2750°C)/[2 × 10⁴ W/m²-K(540°C - 150°C)]
t = -378 W/m-K(-2210°C)/[2 × 10⁴ W/m²-K(390°C)]
t = 835380 W/m/780 × 10⁴ W/m²
t = 835380 W/m/7800000 W/m²
t = 0.1071 m
t = 10.71 cm
So, the maximum thickness of the copper nozzle is 10.71 cm
<h3>b. Maximum thickness for steel nozzle</h3>
Given that for steel
- T = 980°C and
- k = 23.2 W/m-K
Substituting the values of the variables into t, we have
t = -k(T - T")/h(T - T')
t = -23.2 W/m-K(980°C - 2750°C)/[2 × 10⁴ W/m²-K(980°C - 150°C)]
t = -23.2 W/m-K(-1770°C)/[2 × 10⁴ W/m²-K(590°C)]
t = 41064 W/m/1180 × 10⁴ W/m²
t = 41064 W/m/11800000 W/m²
t = 0.00348 m
t = 0.348 cm
t ≅ 0.35 cm
So, the maximum thickness of the steel nozzle is 0.35 cm
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