Question

A thin, square flat plate has $1.2\mathrm{m}$ on each side. Air at ${10}^{\circ }\mathrm{C}$ flows over the top and bottom surfaces of a very rough plate in a direction parallel to one edge, with a velocity of $48\mathrm{m}/\mathrm{s}$. The surface of the plate is maintained at a constant temperature of ${54}^{\circ }\mathrm{C}$. The plate is mounted on a scale that measures a drag force of $1.5\mathrm{N}$. Determine the total heat transfer rate from the plate to the air.

Short answer

The total heat transfer rate from the plate to the air is approximately 503 W.

Step-by-step solution

Calculate the Reynolds number

In order to determine if the flow is laminar or turbulent, we should first calculate the Reynolds number using the following equation: $$Re=\frac{\rho VL}{\mu }$$ Where: - $Re$ is the Reynolds number, - $\rho$ is the density of the fluid (air), - $V$ is the velocity of the fluid, - $L$ is the characteristic length of the object (in our case, the side length of the square plate), - $\mu$ is the dynamic viscosity of the fluid. According to the problem statement, we have $L=1.2\mathrm{m}$ and $V=48\mathrm{m}/\mathrm{s}$. At ${10}^{\circ }\mathrm{C}$, we can approximate the density and dynamic viscosity of air as $\rho =1.25\mathrm{kg}/{\mathrm{m}}^3$ and $\mu =1.8\times {10}^{-5}\mathrm{Pa}\cdot \mathrm{s}$. With these values, we can calculate the Reynolds number: $$Re=\frac{(1.25\mathrm{kg}/{\mathrm{m}}^3)(48\mathrm{m}/\mathrm{s})(1.2\mathrm{m})}{1.8\times {10}^{-5}\mathrm{Pa}\cdot \mathrm{s}}\approx 4\times {10}^5$$ The Reynolds number $4\times {10}^5$ indicates that the flow is turbulent.

Find the Nusselt number

Now we will find the Nusselt number for the flat plate, which relates the convective heat transfer coefficient and the thermal conductivity of the fluid. Since the flow is turbulent, we will use a correlation for a turbulent boundary layer, given by: $$Nu=0.0296R{e}^{\frac{4}{5}}P{r}^{\frac{1}{3}}$$ Where: - $Nu$ is the Nusselt number, - $Re$ is the Reynolds number, - $Pr$ is the Prandtl number. The Prandtl number for air at ${10}^{\circ }\mathrm{C}$ is approximately $Pr=0.71$. Thus, we can calculate the Nusselt number as: $$Nu=0.0296(4\times {10}^5{)}^{\frac{4}{5}}(0.71{)}^{\frac{1}{3}}\approx 363.4$$

Compute the convective heat transfer coefficient

The convective heat transfer coefficient ($h$) can be found using the Nusselt number, given as: $$h=\frac{Nu\cdot k}{L}$$ Where: - $h$ is the convective heat transfer coefficient, - $Nu$ is the Nusselt number, - $k$ is the thermal conductivity of the fluid, - $L$ is the characteristic length. For air at ${10}^{\circ }\mathrm{C}$, the thermal conductivity is approximately $k=0.0261\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})$. Using this value, we can find the convective heat transfer coefficient: $$h=\frac{(363.4)(0.0261\mathrm{W}/(\mathrm{m}\cdot \mathrm{K}))}{1.2\mathrm{m}}\approx 7.92\mathrm{W}/({\mathrm{m}}^2\cdot \mathrm{K})$$

Determine the total heat transfer rate

Finally, we can find the total heat transfer rate ($\dot{Q}$) from the plate to the air using the equation: $$\dot{Q}=hA\mathrm{\Delta }T$$ Where: - $\dot{Q}$ is the total heat transfer rate, - $h$ is the convective heat transfer coefficient, - $A$ is the surface area of the plate, - $\mathrm{\Delta }T$ is the temperature difference between the plate and the air. The surface area of the plate is $A=(1.2\mathrm{m}{)}^2$, and the temperature difference is $\mathrm{\Delta }T=T_{plate}-T_{air}=({54}^{\circ }\mathrm{C}-{10}^{\circ }\mathrm{C})={44}^{\circ }\mathrm{C}$. Now we can find the total heat transfer rate: $$\dot{Q}=(7.92\mathrm{W}/({\mathrm{m}}^2\cdot \mathrm{K}))(1.44{\mathrm{m}}^2)({44}^{\circ }\mathrm{C})\approx 503\mathrm{W}$$ Thus, the total heat transfer rate from the plate to the air is approximately $503\mathrm{W}$.

Key concepts

Reynolds Number

The Reynolds number is a dimensionless quantity used to predict the flow pattern of a fluid in different fluid flow situations. In essence, it helps determine whether the flow will be laminar or turbulent. To calculate the Reynolds number ($Re$), the formula one uses is:$$Re=\frac{\rho VL}{\mu }$$Where:

• $\rho$ represents the density of the fluid, such as air.
• $V$ is the velocity at which the fluid moves.
• $L$ is the characteristic length, which could be the length or diameter in the context of the object involved.
• $\mu$ is the dynamic viscosity of the fluid.

For example, in a scenario where air is blowing over a flat plate, if $Re$ exceeds approximately 4000, it suggests that the flow is turbulent. Understanding the Reynolds number is crucial since it influences the calculation of other quantities like the Nusselt number.

Nusselt Number

The Nusselt number is another dimensionless number central to heat transfer calculations. It represents the ratio of convective to conductive heat transfer across a boundary. For this purpose, it provides an idea of the effectiveness of heat transfer in a fluid. The formula to find the Nusselt number ($Nu$) in turbulent flows is:$$Nu=0.0296R{e}^{\frac{4}{5}}P{r}^{\frac{1}{3}}$$Here:

• $Re$ is the Reynolds number, reflecting the flow's nature.
• $Pr$ stands for the Prandtl number, which will be covered shortly.

The Nusselt number ties into the concept of the convective heat transfer coefficient. A higher Nusselt number indicates more efficient heat transfer. Thus, in designs requiring efficient thermal transfer such as exchangers, engineers aim to maximize this number.

Convective Heat Transfer Coefficient

The convective heat transfer coefficient, denoted as $h$, is a measure of the convective heat transfer from a solid surface to a fluid or vice versa. It determines the rate of heat transfer through convection. The relation involving the Nusselt number is:$$h=\frac{Nu\cdot k}{L}$$Here:

• $Nu$ is the Nusselt number.
• $k$ is the thermal conductivity of the fluid.
• $L$ again is the characteristic length.

The convective heat transfer coefficient changes based on the nature of the flow (laminar or turbulent) and the thermal conditions. It is a vital parameter in determining the heat transfer rate, influencing the design and performance of heating and cooling systems.

Turbulent Flow

Turbulent flow is a type of fluid (gas or liquid) movement characterized by chaotic property changes. This includes rapid variation of pressure and velocity in space and time. Unlike laminar flow, turbulent flow doesn't follow smooth streamlines; instead, it makes eddies and swirls. Determining whether a flow is turbulent can be done using the Reynolds number. If $Re$ is above a certain threshold (commonly 4000 for internal flows), the flow is generally considered turbulent.

In practical applications, turbulent flow increases mixing and, consequently, heat transfer rates; however, it also tends to introduce complexity in design and calculation. The presence of turbulence is advantageous in situations demanding efficient heat exchange, such as in heat exchangers and aerodynamics.

Prandtl Number

The Prandtl number ($Pr$) is a dimensionless number that provides insight into the relative thickness of the momentum and thermal boundary layers of flowing fluids. The formula for calculating the Prandtl number is:$$Pr=\frac{\mu c_p}{k}$$Where:

• $\mu$ is the dynamic viscosity.
• $c_p$ is the specific heat capacity at constant pressure.
• $k$ is the thermal conductivity.

A high Prandtl number implies that momentum diffusivity dominates over thermal diffusivity, meaning the fluid heats up slowly. Conversely, a low Prandtl number indicates heat diffuses faster than momentum. The Prandtl number doesn't vary much for gases at a fixed pressure and temperature but is crucial in analyzing heat transfer processes and ensuring optimal thermal management.