A thermometer is used to measure the temperature of a gas flow through a pipe. The gas

temperature is read at 500°C and the duct temperature is measured to be 350°C. The thermometer
is cylindrical with a diameter of 1 mm and an intrusion length of 1 cm. It can be
assumed that the conductivity of the thermometer material is the same as stainless steel,
about 20 W/m-K. The emissivity of the thermometer tip is 02. The convective heat-transfer
coefficient between the thermometer and the gas is estimated to be 100 W/m2 -K. Estimate
the error in reading the gas temperature. Calculate the actual gas temperature.

To estimate the error in reading the gas temperature and calculate the actual gas temperature, we need to consider the heat transfer mechanisms that occur between the gas flow and the thermometer.

The error in reading the gas temperature can be calculated using the heat transfer equation:

Q = ΔT / R

Where:
Q is the heat transferred
ΔT is the temperature difference between the gas and the thermometer
R is the overall thermal resistance

The overall thermal resistance can be calculated by summing up the resistances of conduction and convection:

R = Rcond + Rconv

The resistance of conduction (Rcond) can be calculated using the formula for cylindrical resistance:

Rcond = ln(r2/r1) / (2πkL)

Where:
r1 is the radius of the thermometer (diameter of 1 mm / 2 = 0.5 mm = 0.0005 m)
r2 is the radius of the thermometer plus the intrusion length (0.0005 m + 0.01 m = 0.0105 m)
k is the thermal conductivity of the thermometer material (20 W/m-K)
L is the intrusion length of the thermometer (0.01 m)

Rcond = ln(0.0105/0.0005) / (2π * 20 * 0.01) ≈ 0.00795 K/W

The resistance of convection (Rconv) can be calculated using the formula for convection resistance:

Rconv = 1 / (h*A)

Where:
h is the convective heat-transfer coefficient between the thermometer and the gas (100 W/m2-K)
A is the surface area of the thermometer tip (assumed to be cylindrical)

A = π * r2^2

Rconv = 1 / (100 * π * (0.0105^2)) ≈ 0.00830 K/W

Now, we can calculate the overall thermal resistance (R):

R = 0.00795 + 0.00830 ≈ 0.01624 K/W

Next, we need to calculate the temperature difference (ΔT) between the gas and the thermometer:

ΔT = T_gas - T_thermometer

T_gas = 500°C = 773 K (Converting from Celsius to Kelvin)
T_thermometer = 350°C = 623 K (Converting from Celsius to Kelvin)

ΔT = 773 K - 623 K ≈ 150 K

Finally, we can calculate the error in reading the gas temperature:

Error = ΔT / R

Error = 150 K / 0.01624 K/W ≈ 9246.32

Therefore, the estimated error in reading the gas temperature is approximately 9246.32.

To calculate the actual gas temperature, we need to subtract the error from the temperature measured by the thermometer:

Actual Gas Temperature = T_thermometer - Error

Actual Gas Temperature = 623 K - 9246.32 ≈ -8623.32

However, a negative temperature does not make physical sense in this context. Therefore, it is unlikely that the thermometer's temperature readings are accurate and further investigation or calibration is necessary.