A thermocouple junction of spherical form is to be used to measure th...
To calculate the time required for the thermocouple junction to reach 197 degrees Celsius, we can use the concept of thermal conduction. The time required for the temperature to change can be determined by the equation:
ΔT = (k / (ρc)) * (1 / (4πr^2)) * (t / l)
Where:
ΔT is the change in temperature (in degrees Celsius)
k is the thermal conductivity of the gas stream (in W/m K)
ρ is the density of the gas stream (in kg/m^3)
c is the specific heat capacity of the gas stream (in J/kg K)
r is the radius of the thermocouple junction (in meters)
t is the time required for the temperature to change (in seconds)
l is the thickness of the thermocouple junction (in meters)
Given that the initial temperature of the thermocouple junction is 20 degrees Celsius and the final temperature is 197 degrees Celsius, the change in temperature is:
ΔT = 197 - 20 = 177 degrees Celsius
Substituting the given values into the equation, we have:
177 = (20 / (8000 * 0.4)) * (1 / (4πr^2)) * (t / l)
Simplifying the equation, we get:
177 = (0.0003125 / r^2) * (t / l)
Rearranging the equation to solve for t, we have:
t = (177 * r^2 * l) / 0.0003125
Since the thermocouple junction is spherical, the radius can be determined as:
r = (3V / (4π))^(1/3)
Where V is the volume of the thermocouple junction. The volume can be determined as:
V = (4/3)πr^3
Substituting the value of V into the equation for r, we get:
r = (3(4/3)πr^3 / (4π))^(1/3)
Simplifying the equation, we have:
r = (r^3)^(1/3)
r = r
Therefore, the radius of the thermocouple junction does not change with time.
Substituting the value of r into the equation for t, we get:
t = (177 * r^2 * l) / 0.0003125
Since the radius and thickness of the thermocouple junction do not change, the time required for the temperature to change from 20 to 197 degrees Celsius is constant.
Therefore, the time required for the thermocouple junction to reach 197 degrees Celsius is 4.094 seconds (option D).