Question

Consider a vertical plate with length \(L\), placed in quiescent air. If the film temperature is \(20^{\circ} \mathrm{C}\) and the average Nusselt number in natural convection is of the form \(\mathrm{Nu}=\mathrm{CRa}_{L}^{n}\), show that the average heat transfer coefficient can be expressed as $$ \begin{aligned} h &=1.51(\Delta T / L)^{1 / 4} \quad 10^{4}<\mathrm{Ra}_{L}<10^{9} \\ h &=1.19 \Delta T^{1 / 3} \quad 10^{10}<\mathrm{Ra}_{L}<10^{13} \end{aligned} $$

Short answer

Question: Show that the average heat transfer coefficient (h) can be expressed in two different ways depending on the range of the Rayleigh number (Ra_L) for a vertical plate with length L in quiescent air at a film temperature of 20°C. Answer: The average heat transfer coefficient (h) can be expressed as: 1. h = 1.51(ΔT / L)^(1/4) for the range 10^4 < Ra_L < 10^9 2. h = 1.19ΔT^(1/3) for the range 10^10 < Ra_L < 10^13

Step-by-step solution

Write the relationship between Nusselt number, heat transfer coefficient, and length

The Nusselt number (Nu) for a vertical plate is related to the heat transfer coefficient (h) and the length of the plate (L) by the following expression: Nu = hL/k where k is the thermal conductivity of the fluid (air in this case).

Express the heat transfer coefficient in terms of Nusselt number and thermal conductivity

Since we want to find an expression for h, we rearrange the equation from Step 1: h = Nu * k / L

Substitute the given expression of Nusselt number in terms of Rayleigh number

We are given the Nusselt number expression Nu = CRa_L^n. Substitute this expression into the equation found in Step 2: h = (CRa_L^n) * k / L

Analyze the given ranges of Rayleigh number

We need to show two different expressions for h, based on the given Rayleigh number (Ra_L) ranges. Let's analyze each range separately: 1. 10^4 < Ra_L < 10^9: In this range, we need to show that h = 1.51(ΔT / L)^(1/4). 2. 10^10 < Ra_L < 10^13: In this range, we need to show that h = 1.19ΔT^(1/3).

Show the expression for the first range of Rayleigh number

For the range 10^4 < Ra_L < 10^9, we have the following relationship between the Nusselt number and the Rayleigh number: Nu = CRa_L^n, where C = 1.51 and n = 1/4 Now, we substitute this into the equation for h: h = (1.51 * Ra_L^(1/4)) * k / L We know that Ra_L = (g * β * ΔT * L^3) / (ν * α), where g is the acceleration due to gravity, β is the thermal expansion coefficient, ΔT is the temperature difference, ν is the kinematic viscosity, and α is the thermal diffusivity. Now we substitute Ra_L in the equation for h: h = (1.51 * ((g * β * ΔT * L^3) / (ν * α))^(1/4)) * k / L After simplification, we get: h = 1.51(ΔT / L)^(1/4)

Show the expression for the second range of Rayleigh number

For the range 10^10 < Ra_L < 10^13, we have the following relationship between the Nusselt number and the Rayleigh number: Nu = CRa_L^n, where C = 1.19 and n = 1/3 Now, we substitute this into the equation for h: h = (1.19 * Ra_L^(1/3)) * k / L We know that Ra_L = (g * β * ΔT * L^3) / (ν * α), where g is the acceleration due to gravity, β is the thermal expansion coefficient, ΔT is the temperature difference, ν is the kinematic viscosity, and α is the thermal diffusivity. Now we substitute Ra_L in the equation for h: h = (1.19 * ((g * β * ΔT * L^3) / (ν * α))^(1/3)) * k / L After simplification, we get: h = 1.19ΔT^(1/3) Both expressions for h are derived, which proves the given expressions for the average heat transfer coefficient depending on the range of the Rayleigh number.