Question

A 2-cm-diameter plastic rod has a thermocouple inserted to measure temperature at the center of the rod. The plastic rod $\left(\rho=1190 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=1465 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and \(k=0.19\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) ) was initially heated to a uniform temperature of \(70^{\circ} \mathrm{C}\) and allowed to be cooled in ambient air at \(25^{\circ} \mathrm{C}\). After \(1388 \mathrm{~s}\) of cooling, the thermocouple measured the temperature at the center of the rod to be \(30^{\circ} \mathrm{C}\). Determine the convection heat transfer coefficient for this process. Solve this problem using the analytical one-term approximation method.

Short answer

Question: Determine the convection heat transfer coefficient for a plastic rod that has been heated and is cooling in ambient air. Short Answer: Using the given information and the analytical one-term approximation method, we can calculate the dimensionless Biot Number and dimensionless time. Solving the equation θ(t) = 1 - 0.75 * (τ·exp(-Bi²·τ)) for the Biot Number, we can then use the Biot number and the given values of Lc and k to find the convection heat transfer coefficient, h, for this process.

Step-by-step solution

Obtain the given information

The diameter of the rod, D = 2 cm = 0.02 m, the density of the plastic rod, ρ = 1190 kg/m³, specific heat capacity at constant pressure, cp = 1465 J/(kg·K), thermal conductivity, k = 0.19 W/(m·K), initial temperature Ti = 70°C, final temperature Tf = 30°C, the ambient temperature Ta = 25°C, and the cooling time t = 1388 s.

Calculate the dimensionless Biot Number

To calculate the Biot number, we need the radius of the rod, r = D/2 = 0.01 m. The characteristic length for a cylinder is Lc = r/2 = 0.005 m. Now, we can calculate the Biot number, Bi = (h·Lc)/k, where h is the convection heat transfer coefficient. We will solve for h later.

Calculate the dimensionless time

First, we need to calculate the thermal diffusivity, α = k/(ρ·cp) = 0.19 W·m/K / (1190 kg/m³ · 1465 J/kg·K). Next, we calculate the dimensionless time, τ = (α·t) / (Lc²) = (α·1388 s) / (0.005 m)².

Find the analytical one-term approximation

We use the analytical one-term approximation for cylinders, given by: θ(t) = (T(t) - Ta) / (Ti - Ta) = 1 - 0.75 * (τ·exp(-Bi²·τ)) where θ(t) is the dimensionless temperature. The final temperature T(t) is 30°C, so substitute this into the formula to find θ(t): θ(t) = (30°C - 25 °C) / (70°C - 25°C)

Solve for the Biot Number

Now, we will substitute the values of θ(t), τ, and the analytical one-term approximation into the equation and then solve for the Bi: θ(t) = 1 - 0.75 * (τ·exp(-Bi²·τ)) Now we solve this equation for Bi.

Calculate the convection heat transfer coefficient

We can now use the Biot number to find the convection heat transfer coefficient, h: Bi = (h·Lc)/k Solve this equation for h using the calculated Bi and given values of Lc and k. After finding the value of h, we have successfully determined the convection heat transfer coefficient for this process.