Question

Airstream at 1 atm flows, with a velocity of $15\mathrm{m}/\mathrm{s}$, in parallel over a 3-m-long flat plate where there is an unheated starting length of $1\mathrm{m}$. The airstream has a temperature of ${20}^{\circ }\mathrm{C}$ and the heated section of the flat plate is maintained at a constant temperature of ${80}^{\circ }\mathrm{C}$. Determine $(a)$ the local convection heat transfer coefficient at the trailing edge and (b) the average convection heat transfer coefficient for the heated section.

Short answer

Question: Calculate (a) the local convection heat transfer coefficient at the trailing edge and (b) the average convection heat transfer coefficient for the heated section of a flat plate given the following information: Flow velocity = 10 m/s, Length of the flat plate = 3 m, Unheated starting length = 0.5 m, Airstream temperature = 20°C, Heated section temperature = 60°C. Use the Blasius correlation for laminar flow over a flat plate. Answer: (a) The local convection heat transfer coefficient at the trailing edge can be calculated as follows: 1. Calculate the mean temperature: $T_m=\frac{20+60}{2}=40$°C 2. Determine the air properties at 40°C: $\rho$, $\mu$, $k$, $C_p$, and $\nu$ (refer to air property tables or an online calculator) 3. Calculate the Reynolds number at the trailing edge: $R{e}_x=\frac{U_{\mathrm{\infty }}x}{\nu }$ 4. Determine the Nusselt number at the trailing edge using the Blasius correlation: $N{u}_x=0.332R{e}_x^{1/2}P{r}^{1/3}$ 5. Calculate the local convection heat transfer coefficient at the trailing edge: $h_x=\frac{k}{x}N{u}_x$ (b) The average convection heat transfer coefficient for the heated section can be calculated as follows: 1. Calculate the average Nusselt number for the heated section using the Petukhov correlation: $N{u}_L=CR{e}_L^{m}P{r}^n\frac{1}{{[1+L_0/L]}^{0.75}}$ 2. Calculate the average convection heat transfer coefficient for the heated section: $h_L=\frac{k}{L}N{u}_L$ Remember to use consistent units and the appropriate air properties when performing the calculations.

Step-by-step solution

(Step 1: Calculate the mean temperature of airstream and heated section)

(First, find the mean temperature by averaging the given airstream temperature $T_{\mathrm{\infty }}$ and heated section temperature $T_s$. Mean temperature, $T_m=\frac{T_{\mathrm{\infty }}+T_s}{2}$)

(Step 2: Determine the properties of air at the mean temperature)

(Using the mean temperature, find the properties of air like density $\rho$, dynamic viscosity $\mu$, thermal conductivity $k$, specific heat $C_p$, and kinematic viscosity $\nu$. You can find these properties in the air property tables or use an online calculator for the same. Remember to use consistent units.)

(Step 3: Calculate the Reynolds number at the trailing edge)

(Next, calculate the Reynolds number at the trailing edge, which is at a distance of 3 meters (total length of the flat plate) from the leading edge. The Reynolds number, $R{e}_x=\frac{U_{\mathrm{\infty }}x}{\nu }$ Where $U_{\mathrm{\infty }}$ is the flow velocity, $x$ is the distance from the leading edge (3 m in this case) and $\nu$ is the kinematic viscosity.)

(Step 4: Determine the Nusselt number at the trailing edge using correlation)

(With the Reynolds number calculated, now we can determine the Nusselt number ($N{u}_x$) at the trailing edge using an appropriate correlation. Since we have an unheated starting length, the unheated length should be subtracted from the total length before using the correlation. In this case, you can use the Blasius correlation for laminar flow over a flat plate: $N{u}_x=0.332R{e}_x^{1/2}P{r}^{1/3}$ Where $Pr$ is the Prandtl number given by $Pr=\frac{C_p\mu }{k}$.)

(Step 5: Calculate the local convection heat transfer coefficient at the trailing edge)

(Now we can calculate the local convection heat transfer coefficient, $h_x$, at the trailing edge. $h_x=\frac{k}{x}N{u}_x$)

(Step 6: Calculate the average Nusselt number for the heated section)

(Next, we will calculate the average Nusselt number, $N{u}_L$, for the heated section using Petukhov correlation, which accounts for the unheated starting length ($L_0$) and heated length $L$. $N{u}_L=CR{e}_L^{m}P{r}^n\frac{1}{{[1+L_0/L]}^{0.75}}$ Where $R{e}_L=\frac{U_{\mathrm{\infty }}L}{\nu }$, and $C$, $m$, and $n$ are constants depending on the flow type (for laminar flow, $C=0.036$, $m=0.8$, and $n=0.33$).)

(Step 7: Calculate the average convection heat transfer coefficient for the heated section)

(Finally, we can calculate the average convection heat transfer coefficient, $h_L$, for the heated section. $h_L=\frac{k}{L}N{u}_L$) Now, we have successfully found both (a) the local convection heat transfer coefficient at the trailing edge and (b) the average convection heat transfer coefficient for the heated section.

Key concepts

Reynolds Number

The Reynolds Number is a critical concept in fluid dynamics that helps determine the flow characteristics of a fluid. In simpler terms, it tells us whether the flow is laminar or turbulent. The Reynolds Number is a dimensionless quantity and is calculated using the formula:

• $$R{e}_x=\frac{U_{\mathrm{\infty }}x}{u}$$

Here, $U_{\mathrm{\infty }}$ is the velocity of the flow, $x$ is the distance from the leading edge, and $u$ is the kinematic viscosity.
The importance of the Reynolds Number comes into play when analyzing the flow over objects like the flat plate in our problem. It helps to decide which correlations to use for calculating heat transfer characteristics. Understanding if the flow is laminar or turbulent affects how we approach the calculation of heat transfer coefficients, a crucial part of such exercises.

Nusselt Number

In the context of convection heat transfer, the Nusselt Number is particularly important because it relates the convective to conductive heat transfer across a boundary. This dimensionless number helps us understand the efficiency of heat transfer from the surface to the fluid.

• For a flat plate experiencing laminar flow, the Nusselt Number can be derived using correlations like the Blasius equation.
• The formula used in our context is:$$N{u}_x=0.332R{e}_x^{1/2}P{r}^{1/3}$$

This correlation considers the Reynolds Number and the Prandtl Number, and it allows for calculating the local convective heat transfer coefficient.
In summary, the Nusselt Number helps in quantifying the increased thermal energy transfer as a result of convection compared to conduction alone.

Heat Transfer Coefficient

The Heat Transfer Coefficient, denoted usually as $h$, is a key component in quantifying the rate of heat transfer by convection. It is defined as the heat transfer rate per unit area and per unit temperature difference.

• The local convection heat transfer coefficient $h_x$ at the trailing edge can be calculated using:$$h_x=\frac{k}{x}N{u}_x$$
• Here, $k$ is the thermal conductivity, $x$ is the distance from the leading edge, and $N{u}_x$ is the local Nusselt Number.

Critical for design and analysis in heat transfer applications, the heat transfer coefficient indicates how effectively heat is being transferred in different scenarios. For this problem, finding the local and average coefficients at various points on the plate can inform design considerations for controlling temperatures in engineering applications.

Laminar Flow

Laminar Flow describes a type of fluid flow where the fluid moves in smooth layers or laminae, with little to no mixing between the layers. In terms of velocity, the fluid flows in parallel lines.

• It is typically characterized by low Reynolds Numbers ($Re<2000$), indicating a stable and uniform flow.
• In this exercise, the initial assumptions guide us to consider laminar flow given the Reynolds Number and properties of the air over the flat plate.

The significance of confirming laminar flow lies in its impact on calculating the heat transfer rates effectively. The approach to determining Nusselt Numbers and necessary correlations differ if the flow is turbulent, thus simplifying our calculations when assuming laminar flow.

Prandtl Number

The Prandtl Number is another dimensionless quantity that is essential in the analysis of heat transfer problems. It relates the momentum diffusivity (viscous diffusion) to thermal diffusivity.

• The Prandtl Number is expressed as:$$Pr=\frac{C_p\mu }{k}$$
• Where $C_p$ is the specific heat, $\mu$ is the dynamic viscosity, and $k$ is the thermal conductivity.

Understanding the Prandtl Number helps in determining the thermal boundary layer's characteristics and interacting with the flow's velocity profile.
In practical terms, it aids in using the right correlations to find the Nusselt Number, which in turn allows for an accurate calculation of the convection heat transfer coefficients. By knowing the Prandtl Number, students can better gauge how energy is being conducted versus convected, highlighting its key role in fluid flow and heat transfer analysis.