Question

Consider a 2-ft \(\times 2-\mathrm{ft}\) thin square plate in a room at \(75^{\circ} \mathrm{F}\). One side of the plate is maintained at a temperature of \(130^{\circ} \mathrm{F}\), while the other side is insulated. Determine the rate of heat transfer from the plate by natural convection if the plate is ( \(a\) ) vertical, \((b)\) horizontal with hot surface facing up, and (c) horizontal with hot surface facing down.

Short answer

Question: Calculate the rate of heat transfer for a thin square plate in a room in three different orientations: a) vertically oriented, b) horizontally oriented with the hot surface facing up, and c) horizontally oriented with the hot surface facing down. The plate is 2ft x 2ft, and the temperature difference between the plate and the surrounding air is ΔT. Answer: To calculate the rate of heat transfer for each orientation, follow these steps: 1. Calculate the Grashof (Gr) and Prandtl (Pr) numbers. 2. Calculate the Nusselt numbers (Nu) for each case: a) vertical orientation, b) horizontal orientation with the hot surface facing up, c) horizontal orientation with the hot surface facing down. 3. Calculate heat transfer coefficients (h) for each case. 4. Calculate the rate of heat transfer (Q) for each case using the expression Q = h * A * ΔT, where A is the surface area of the plate. Follow the mentioned steps using the provided formulae and properties of the air to determine the rate of heat transfer for each orientation.

Step-by-step solution

Calculate Grashof and Prandtl numbers

Calculate the Grashof and Prandtl numbers, which are dimensionless numbers essential for natural convection heat transfer calculations. The Grashof number (Gr) is given by the formula: Gr = (g * β * ΔT * L^3) / ν^2 where g is the acceleration due to gravity (9.81 m/s^2), β is the coefficient of thermal expansion (approximately equal to 1/T_mean, where T_mean is the average temperature in Kelvin), ΔT is the temperature difference between the plate and the surrounding air, L is the characteristic length (2ft converted to meters), and ν is the kinematic viscosity of air. The Prandtl number (Pr) is determined as: Pr = ν / α where α is the thermal diffusivity of the air.

Calculate Nusselt numbers

Calculate the Nusselt numbers (Nu) for each case using the Grashof and Prandtl numbers: a) For a vertical plate, use the Churchill-Chu correlation: Nu = 0.825 + 0.387(Gr * Pr)^{1/6} / {[(0.96 + 3.44 / Pr)^{2/3} + (0.387 / {(Gr * Pr)^{1/9}})^3}]^{1/4}} b) For a horizontal plate with hot surface facing up, use the Morgan correlation: Nu = 0.54(Gr * Pr)^{1/4} c) For a horizontal plate with hot surface facing down, use the Churchill and Chu correlation: Nu = 0.27(Gr * Pr)^{1/4}

Calculate heat transfer coefficients

Calculate heat transfer coefficients (h) for each case by using the Nusselt numbers and the thermal conductivity of air (k). The heat transfer coefficient is given by: h = k * (Nu) / L

Calculate the rate of heat transfer

Calculate the rate of heat transfer (Q) for each case using the expression: Q = h * A * ΔT where A is the surface area of the plate (2ft * 2ft converted to square meters) and ΔT is the temperature difference between the plate and the surrounding air. By following these steps, you can determine the rate of heat transfer by natural convection based on the orientation of the plate and whether the hot surface is facing up or down.