Question

Hydrogen gas at $1\mathrm{atm}$ is flowing in parallel over the upper and lower surfaces of a 3-m-long flat plate at a velocity of $2.5\mathrm{m}/\mathrm{s}$. The gas temperature is ${120}^{\circ }\mathrm{C}$ and the surface temperature of the plate is maintained at ${30}^{\circ }\mathrm{C}$. Using the EES (or other) software, investigate the local convection heat transfer coefficient and the local total convection heat flux along the plate. By varying the location along the plate for $0.2\le x\le 3\mathrm{m}$, plot the local convection heat transfer coefficient and the local total convection heat flux as functions of $x$. Assume flow is laminar but make sure to verify this assumption. 7-31 Carbon dioxide and hydrogen as ideal gases at $1\mathrm{atm}$ and $-{20}^{\circ }\mathrm{C}$ flow in parallel over a flat plate. The flow velocity of each gas is $1\mathrm{m}/\mathrm{s}$ and the surface temperature of the 3 -m-long plate is maintained at ${20}^{\circ }\mathrm{C}$. Using the EES (or other) software, evaluate the local Reynolds number, the local Nusselt number, and the local convection heat transfer coefficient along the plate for each gas. By varying the location along the plate for $0.2\le x\le 3\mathrm{m}$, plot the local Reynolds number, the local Nusselt number, and the local convection heat transfer coefficient for each gas as functions of $x$. Discuss which gas has higher local Nusselt number and which gas has higher convection heat transfer coefficient along the plate. Assume flow is laminar but make sure to verify this assumption.

Short answer

Question: Calculate the local convection heat transfer coefficient and local total convection heat flux along a 3m-long flat plate for hydrogen gas flowing in parallel at specific conditions, and plot them as functions of x, considering laminar flow. Also, evaluate the local Reynolds number, the local Nusselt number, and the local convection heat transfer coefficient for carbon dioxide and hydrogen gases with given conditions, and plot them as functions of x. Discuss which gas has a higher local Nusselt number and convection heat transfer coefficient.

Step-by-step solution

1. Calculations for Hydrogen gas

First, we need to calculate the local convection heat transfer coefficient by knowing basic properties such as thermal conductivity, dynamic viscosity, and heat capacity of hydrogen gas at given temperatures. It's necessary to determine the flow regime (laminar or turbulent) by calculating the Reynolds number. Reynolds number, $Re=\frac{V\cdot x}{\nu }$ If the Reynolds number is less than 5 x 10^5 (Re < 5 x 10^5), we can consider the flow to be laminar. For laminar flow, we can use the following relation to compute the local Nusselt number: Local Nusselt number, $Nu=0.332R{e}^{\frac{1}{2}}P{r}^{\frac{1}{3}}$ The local convection heat transfer coefficient can be calculated using: Local convection heat transfer coefficient, $h=\frac{k\cdot Nu}{x}$ Once we have the local convection heat transfer coefficient, we can use it to calculate the local total convection heat flux: $Localconvectionheatflux=h\cdot A\cdot \mathrm{\Delta }T$ where $A=width\cdot x$ and $\mathrm{\Delta }T=T_{\text{surface}}-T_{\text{gas}}$.

2. Plotting Hydrogen gas results

Next, we need to plot these quantities as functions of x for hydrogen gas. We should vary the location along the plate ($0.2\le x\le 3\mathrm{m}$) and create a plot for the local convection heat transfer coefficient and local total convection heat flux.

3. Calculations for Carbon Dioxide and Hydrogen gases

Next, we need to repeat the calculations (Reynolds number, Nusselt number, and convection heat transfer coefficient) for carbon dioxide and hydrogen gases at the given conditions. We need to determine and verify if the flow is laminar for both gases, and again, we have to use the laminar flow equation for the local Nusselt number.

4. Plotting CO₂ and H₂ results

Once we have the results for carbon dioxide and hydrogen gases, we need to plot the local Reynolds number, local Nusselt number, and local convection heat transfer coefficient as functions of x for each gas.

5. Discussion

Finally, based on the plotted results for both gases, we need to discuss which gas has a higher local Nusselt number and convection heat transfer coefficient along the 3-meter-long flat plate.

Key concepts

Reynolds Number

When dealing with fluid flows over surfaces, the Reynolds number is a fundamental parameter that helps us understand the nature of the flow. The Reynolds number (Re) is calculated as $Re=\frac{V\cdot x}{u}$, where $V$ is the velocity of the fluid, $x$ is the characteristic length (such as the distance along a flat plate), and $u$ is the kinematic viscosity of the fluid.

The Reynolds number essentially tells us whether the flow is laminar or turbulent. For flows over flat plates, a Reynolds number less than 500,000 ($Re<5\times {10}^5$) suggests that the flow is laminar. This information is crucial especially in heat transfer analysis because it affects the heat transfer characteristics.

Understanding whether the flow is laminar or turbulent is the first step in determining how heat will be transferred from the surface to the fluid, and vice versa.

Nusselt Number

The Nusselt number (Nu) is a dimensionless number that is crucial in the study of heat transfer, specifically in convection processes. It relates the rate of convective heat transfer to the rate of conductive heat transfer across the fluid. In simpler terms, it gives us an idea of the efficiency of heat transfer through a fluid.

For laminar flow, the local Nusselt number can be calculated using the formula $Nu=0.332R{e}^{\frac{1}{2}}P{r}^{\frac{1}{3}}$, where $Pr$ is the Prandtl number. The Prandtl number depends on the properties of the fluid, including viscosity and thermal conductivity, and represents the ratio of momentum diffusivity to thermal diffusivity.

Higher Nusselt numbers indicate more efficient convection over the surface, which is instrumental in designing systems for optimal thermal management. It helps engineers to decide appropriate materials and design dimensions for achieving desired thermal outputs.

Laminar Flow

Laminar flow is characterized by smooth and orderly fluid motion, where the fluid moves in parallel layers with minimal mixing between them. This flow regime occurs when the Reynolds number is low, typically below 500,000, as specified for flow over flat plates.

In the context of our exercise, determining if the flow is laminar is a foundational step because it dictates the appropriate equations and methodologies, such as the ones used for calculating the Nusselt number. Laminar flows are often associated with higher resistance to heat and momentum transfer compared to turbulent flows, but their predictability makes them valuable in engineering analysis.

Knowing that the flow is laminar helps in simplifying the heat transfer equations, allowing for more precise calculations of the convection heat transfer coefficients.

Heat Flux Analysis

Heat flux refers to the rate of heat energy transfer through a given surface, typically expressed in watts per square meter (W/m^2). In convection heat transfer, analyzing the heat flux is vital for understanding how much heat is being transferred between the plate and the surrounding fluid.

The local convection heat flux can be determined using the equation $q=h\cdot A\cdot \mathrm{\Delta }T$, where $h$ is the local convection heat transfer coefficient, $A$ is the area through which heat is being transferred, and $\mathrm{\Delta }T$ is the temperature difference between the plate surface and the fluid.

By evaluating heat flux along different sections of the plate, engineers can optimize the thermal performance of a system, ensuring that it functions efficiently. This kind of analysis is crucial, not just for thermodynamics, but also for designing heating and cooling systems, enhancing energy efficiency, and maximizing performance in various engineering applications.