A lagged copper rod has a uniform
cross-sectional area of 1.0cm3 and length 20.0cm.
when steady state is attained, the temperature of one
end of the rod is 120degree Celsius and other end is
0degree Celsius. Calculate; (1)the temperature gradient
(2).rate of heat flow
(3).The temperature at a point 8.0cm from the hot end. (the thermal conductivity of copper =380Wm-1k-1)
area = 1 cm^2 * 10^-4 m^2/cm^2 = 10^-4 m^2
length = 0.20 m
delta T = -120
dT/dx = -120/0.20 = - 600 deg C/m
Q = 380 W m^-1 degC^-1 * 10^-4 m^2 * 600 degC/m
= 380*10^-4*600 Watts
= 228 watts
dT/dx = -600
if x = 8 cm = 0.08 m
T = 120 - 600(.08)
= 72 degC
other way
(8/20)120= 48 deg lost
120 - 48 = 72 deg C luckily
Thanks bro... I appreciate a lot
You are welcome.
To solve this problem, we need to use the formula for the rate of heat flow and the temperature gradient. The rate of heat flow (Q) is calculated using the formula:
Q = (k * A * ΔT) / L
Where:
Q is the rate of heat flow
k is the thermal conductivity of copper
A is the cross-sectional area of the rod
ΔT is the temperature difference between the ends of the rod
L is the length of the rod
Let's calculate the temperature gradient first.
(1) Temperature gradient:
The temperature gradient (dT/dx) gives the change in temperature per unit length along the rod. It can be calculated using the formula:
dT/dx = ΔT / L
where:
ΔT is the temperature difference between the ends of the rod
L is the length of the rod
Given:
ΔT = 120°C - 0°C = 120°C
L = 20.0 cm = 0.20 m
Substituting the given values into the formula:
dT/dx = 120°C / 0.20 m = 600°C/m
So, the temperature gradient is 600°C/m.
(2) Rate of heat flow:
We can now calculate the rate of heat flow using the formula mentioned earlier.
Given:
k = 380 W/m·K
A = 1.0 cm3 = 1.0 × 10^(-6) m³
Substituting the given values:
Q = (k * A * ΔT) / L
= (380 W/m·K * 1.0 × 10^(-6) m³ * 120°C) / 0.20 m
= (380 * 1.0 × 10^(-6) * 120) / 0.20 W
= 0.228 W
So, the rate of heat flow is 0.228 W.
(3) Temperature at a point 8.0 cm from the hot end:
To calculate the temperature at this point, we can use the formula:
T = T_hot - (dT/dx) * x
where:
T is the temperature at the given point
T_hot is the temperature at the hot end of the rod
dT/dx is the temperature gradient
x is the distance from the hot end of the rod
Given:
T_hot = 120°C
dT/dx = 600°C/m
x = 8.0 cm = 0.08 m
Substituting the given values:
T = 120°C - (600°C/m * 0.08 m)
= 120°C - 48°C
= 72°C
So, the temperature at a point 8.0 cm from the hot end is 72°C.