A 1.4-kg aluminum household iron has a 500 W heating element. The surface area is 0.046 m2. The ambient temperature is 21°C and the surface heat transfer coefficient is 11 W/(m2 K). How long after the iron is plugged in will its temperature reach 104°C?

Question

GIVEN

• An aluminum household iron

• Mass of the iron (M) = 1.4 kg

• Power output ( Q

G ) = 500 W

• Surface area (As) = 0.046 m2

• The ambient temperature (T?) = 21°C

• The heat transfer coefficient ( )ch = 11 W/(m2 K)

FIND

• How long after the iron is plugged in will its temperature reach 104°C

ASSUMPTIONS

• Constant heat transfer coefficient

• The mass given is for the heated aluminum portion only

SKETCH

Answer 1

To calculate the Biot number for this problem, we must first calculate the characteristic length



The Biot number is



Therefore, the lumped capacity method may be used. The energy balance for the iron is

Change in internal energy = heat generation – net heat flow from the iron.





Then the heat balance can be written



This is a linear, first order, non-homogeneous differential equation. The solution to the homogeneous equation is ?h = c e–mt and a particular solution is ?p = c. The general solution is the sum of the homogeneous and particular solutions



Substituting this into the heat balance



Applying the initial condition, ? = 0 at t = 0 yields



Therefore, the temperature-time history of the iron is given by