Question

Determine the convection heat transfer coefficient for the flow of \((a)\) air and \((b)\) water at a velocity of \(2 \mathrm{~m} / \mathrm{s}\) in an \(8-\mathrm{cm}\) diameter and 7-m-long tube when the tube is subjected to uniform heat flux from all surfaces. Use fluid properties at \(25^{\circ} \mathrm{C}\).

Short answer

(a) The convection heat transfer coefficient for the flow of air inside the tube is approximately 44.73 W/(m²·K). (b) The convection heat transfer coefficient for the flow of water inside the tube is approximately 43346.84 W/(m²·K).

Step-by-step solution

Obtain fluid properties at 25°C

For air at 25°C: - Density (ρ) = 1.184 kg/m³ - Specific heat (Cp) = 1005 J/(kg·K) - Thermal conductivity (k) = 0.02624 W/(m·K) - Dynamic viscosity (μ) = 1.849x10⁻⁵ kg/(m·s) For water at 25°C: - Density (ρ) = 997 kg/m³ - Specific heat (Cp) = 4182 J/(kg·K) - Thermal conductivity (k) = 0.6075 W/(m·K) - Dynamic viscosity (μ) = 8.9x10⁻⁴ kg/(m·s)

Calculate the Reynolds number (Re)

Re = (ρ * V * D) / μ For air: Re_air = (1.184 * 2 * 0.08) / 1.849x10⁻⁵ = 10237.43 For water: Re_water = (997 * 2 * 0.08) / 8.9x10⁻⁴ = 179328.09

Determine the type of flow (laminar or turbulent)

For both fluids: - If Re < 2300, the flow is laminar - If Re > 2300, the flow is turbulent For air: Re_air = 10237.43, the flow is turbulent For water: Re_water = 179328.09, the flow is turbulent

Use appropriate correlation to calculate Nusselt number (Nu)

Since the flow is turbulent for both fluids, we will use Dittus-Boelter equation, for uniform heat flux: Nu = 0.023 * Re^0.8 * Pr^(1/3) For air: Pr_air = (1.849×10⁻⁵ * 1005) / 0.02624 = 0.707 Nu_air = 0.023 * (10237.43)^0.8 * (0.707)^(1/3) = 137.13 For water: Pr_water = (8.9×10⁻⁴ * 4182) / 0.6075 = 6.108 Nu_water = 0.023 * (179328.09)^0.8 * (6.108)^(1/3) = 5715.27

Calculate the convection heat transfer coefficient (h)

h = (k * Nu) / D For air: h_air = (0.02624 * 137.13) / 0.08 = 44.73 W/(m²·K) For water: h_water = (0.6075 * 5715.27) / 0.08 = 43346.84 W/(m²·K) The convection heat transfer coefficients for the flow of (a) air and (b) water inside the tube are approximately: (a) h_air = 44.73 W/(m²·K) (b) h_water = 43346.84 W/(m²·K)

Key concepts

Reynolds Number

In fluid mechanics, the Reynolds Number is a crucial dimensionless quantity used to predict the flow patterns in different fluid flow situations. It describes the ratio of inertial forces to viscous forces and provides insight into whether the flow will be laminar or turbulent. The formula for calculating the Reynolds Number is given by:\[ \text{Re} = \frac{\rho \cdot V \cdot D}{\mu} \]Where:

• \( \rho \) is the fluid density
• \( V \) is the velocity of the fluid
• \( D \) is the characteristic length, often taken as the diameter in tube flows
• \( \mu \) is the dynamic viscosity

The significance of Reynolds Number lies in its ability to indicate the behavior of flow:

• If \( \text{Re} < 2300 \), the flow is typically considered laminar, meaning it flows in parallel layers with no disruption between them.
• If \( \text{Re} > 2300 \), the flow starts to become turbulent, characterized by random and chaotic changes.

Nusselt Number

The Nusselt Number is another dimensionless number important in heat transfer analysis, specifically in convection. It represents the enhancement of heat transfer through a fluid layer as a result of convection relative to that by conduction. The formula to determine the Nusselt Number in a turbulent flow, using the Dittus-Boelter Equation, is:\[ \text{Nu} = 0.023 \cdot \text{Re}^{0.8} \cdot \text{Pr}^{1/3} \]The Nusselt Number allows engineers to predict the convection heat transfer coefficient from the known properties of the fluid and flow conditions. A higher Nusselt Number indicates a greater enhancement of heat transfer. In the exercise given, knowing the turbulent nature of the flow, the Nusselt Number is calculated using the Dittus-Boelter equation for both air and water.

Dittus-Boelter Equation

The Dittus-Boelter Equation is a commonly used empirical formula in thermal fluid systems to estimate the convective heat transfer coefficient for a fluid flowing through a pipe. It is specifically applied to turbulent flow situations. The formula is written as:\[ \text{Nu} = 0.023 \cdot \text{Re}^{0.8} \cdot \text{Pr}^{0.3} \]This equation helps in understanding how the heat transfer efficiency improves with increasing Reynolds and Prandtl numbers. It takes into account the significance of convection over conduction for different fluid characteristics and flow conditions.

Thermal Conductivity

Thermal Conductivity is a physical property that indicates a material's ability to conduct heat. It's denoted by \( k \) and reflects how well heat is transferred through a material due to a temperature gradient. The units of thermal conductivity are W/(m·K).When considering heat transfer in fluids like air and water, thermal conductivity plays an important role in determining the effectiveness of heat exchange in processes, such as those occurring in heat exchangers and cooling systems. Higher thermal conductivity means better capacity of the material to conduct heat.

Dynamic Viscosity

Dynamic Viscosity is a measure of a fluid's resistance to gradual deformation by shear or tensile stress. It is symbolized by \( \mu \) and is notably measured in kg/(m·s). Dynamic viscosity influences how a fluid flows under force and is an essential factor in determining the Reynolds Number.In fluid dynamic calculations, such as the one described in the exercise, understanding the viscosity is crucial because it affects the flow characterizations as either laminar or turbulent. Higher viscosity means a more significant resistance to flow, impacting how easily the fluid moves through the pipe.