Question

A hot brass plate is having its upper surface cooled by impinging jet of air at temperature of \(15^{\circ} \mathrm{C}\) and convection heat transfer coefficient of \(220 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The \(10-\mathrm{cm}\) thick brass plate \(\left(\rho=8530 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=380 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\), and \(\alpha=33.9 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) has a uniform initial temperature of \(650^{\circ} \mathrm{C}\), and the bottom surface of the plate is insulated. Determine the temperature at the center plane of the brass plate after 3 minutes of cooling. Solve this problem using analytical oneterm approximation method (not the Heisler charts).

Short answer

Question: Calculate the temperature at the center plane of the brass plate after 3 minutes of cooling using the analytical one-term approximation method. Solution: 1. The Biot number was calculated (given the factors like convection heat transfer coefficient and thermal conductivity): Bi = hLc / k 2. The Fourier number was determined (using the time for cooling and the thermal diffusivity): Fo = αt / Lc² 3. The standardized temperature was calculated (with initial temperature and air temperature): θ* ≈ (2exp(-Bi·Fo))/(πBi)·sinh(π√Fo) 4. The temperature at the center plane was found using the formula: T = T∞ + (Ti - T∞)·θ*

Step-by-step solution

Calculate the Biot number

The Biot number (Bi) is a dimensionless quantity that represents the ratio of the convection heat transfer resistance to conduction heat transfer resistance. It's calculated as: Bi = \(\frac{hL_{c}}{k}\), where: \(h = 220 \mathrm{~W}/\mathrm{m}^{2}\cdot\mathrm{K}\) (convection heat transfer coefficient), \(L_{c} = \frac{L}{2} = \frac{10 \times 10^{-2} \mathrm{m}}{2}\) (the characteristic length, which is considered as half the thickness of the plate for this case), \(k = 110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (thermal conductivity of brass).

Calculate the Fourier number

The Fourier number (Fo) is a dimensionless quantity used to analyze transient heat conduction problems. It's calculated as: Fo = \(\frac{\alpha t}{L_{c}^{2}}\), where: \(\alpha = 33.9 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) (thermal diffusivity), \(t = 3 \times 60 \mathrm{~s}\) (time after cooling).

Calculate the standardized temperature

The standardized temperature (\(\theta^{\ast}\)) is used to simplify the temperature calculations and is given by: \(\theta^{\ast}\) = \(\frac{T - T_{\infty}}{T_{i} - T_{\infty}}\), where: \(T\) is the temperature we want to find, \(T_{\infty} = 15^{\circ} \mathrm{C}\) (air temperature), \(T_{i} = 650^{\circ} \mathrm{C}\) (initial plate temperature). Then, \(\theta^{\ast}\) can be calculated using the one-term approximation method as: \(\theta^{\ast} \approx \frac{2\text{exp}\left(-\text{Bi}\cdot\text{Fo}\right)}{\pi \text{Bi}}\cdot\text{sinh}\left(\pi\sqrt{\text{Fo}}\right)\)

Calculate the temperature at the center plane

Now we can solve for the temperature at the center plane using the formula: \(T = T_{\infty} + (T_{i} - T_{\infty})\cdot \theta^{\ast}\) Follow these steps to determine the temperature at the center plane of the brass plate after 3 minutes of cooling using the analytical one-term approximation method.

Key concepts

Biot Number

The Biot number (Bi) is crucial in assessing the heat transfer efficiency between a solid and a fluid. It is dimensionless and is defined as the ratio of conductive to convective heat transfer resistance. This gives insight into the thermal behavior of the material being cooled or heated. For our brass plate, Biot number is calculated using the formula:\[ \text{Bi} = \frac{hL_c}{k} \]where:

• \(h = 220 \, \text{W/m}^2\cdot \text{K}\) is the convection heat transfer coefficient.
• \(L_c\), the characteristic length, is half the thickness of the plate.
• \(k = 110 \, \text{W/m} \cdot \text{K}\) is the thermal conductivity of brass.

This number helps determine if the assumption of uniform temperature inside the material is valid. If Bi is small (typically less than 0.1), the conduction within the solid is rapid compared to convection, leading to a nearly uniform temperature.

Fourier Number

The Fourier number (Fo) is another dimensionless parameter used in the study of transient heat conduction. It indicates the degree to which heat has diffused through a material over a certain time period. The formula for Fourier number is:\[ \text{Fo} = \frac{\alpha t}{L_c^2} \]where:

• \(\alpha = 33.9 \times 10^{-6} \, \text{m}^2/\text{s}\) is the thermal diffusivity of the material.
• \(t = 3 \times 60 \, \text{s}\) represents the time duration.

A higher Fourier number indicates more significant heat diffusion, meaning the material's temperature has had more opportunity to become homogeneous. It's particularly useful in evaluating how quickly heat is diffused within the solid.

Impinging Jet Cooling

Impinging jet cooling involves directing a flow of fluid onto a surface to enhance the rate of convective heat transfer. This cooling method is highly effective in applications requiring rapid heat removal, making it ideal for cooling hot surfaces like our brass plate. The factors affecting this process include:

• The jet’s velocity and temperature.
• The fluid’s properties, like viscosity and specific heat capacity.
• The configuration between the jet and surface, such as angle and distance.

This technique exploits the high convective heat transfer coefficient associated with the impinging jet, resulting in efficient and uniform surface temperature reduction.

Transient Heat Conduction

Transient heat conduction refers to the phenomenon where temperature variations within a body change over time. Unlike steady-state conduction, where temperatures remain constant, transient conduction accommodates changes, such as in our cooling brass plate example. Key points to understand include:

• Thermal conductivity and diffusivity, which dictate how rapidly temperature changes within the material.
• The geometry of the material, which affects the heat distribution.
• Boundary and initial conditions, defining how temperature evolves from initial to subsequent thermal states.

The analytical approach used, like the one-term approximation method, simplifies predicting how internal temperatures of materials adjust during transient states.

One-Term Approximation Method

The one-term approximation method is an analytical solution technique used to estimate temperatures during transient heat conduction problems. This method simplifies complex partial differential equations into a more manageable form, particularly effective for materials with uniform initial temperatures, like our brass plate.The standardized temperature \(\theta^*\) is approximated by:\[ \theta^* \approx \frac{2\exp(-\text{Bi} \cdot \text{Fo})}{\pi \text{Bi}} \cdot \sinh (\pi \sqrt{\text{Fo}}) \]After computing \(\theta^*\), it allows determination of the temperature at any point by:\[ T = T_{\infty} + (T_i - T_{\infty}) \cdot \theta^* \]This approach is especially useful for quick estimations without needing graphical methods like Heisler charts, providing a practical way to analyze heat conduction problems.