3.3 Forced Convection: Internal Flow
6 min read•august 13, 2024
Internal flow in pipes and ducts is crucial for heat transfer in many systems. As fluid moves through confined spaces, velocity and temperature profiles develop, affecting heat exchange. Understanding these processes helps engineers design efficient heating and .
Factors like , friction, and impact internal flow behavior. Correlations for laminar and turbulent flows in various geometries allow us to predict heat transfer rates. This knowledge is essential for optimizing and other thermal management devices.
Internal Flow Characteristics
Fluid Boundary and Velocity Profile Development
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- Internal flow is characterized by the fluid being completely bounded by solid surfaces, such as in pipes or ducts (cylindrical pipes, rectangular ducts)
- The of the fluid in internal flow develops over a certain length, known as the , before becoming fully developed
- In the hydrodynamic entry region, the velocity profile changes from a uniform profile at the inlet to a fully developed profile downstream
- The shape of the fully developed velocity profile depends on the Reynolds number (laminar or )
Thermal Boundary Layer Development and Nusselt Number
- The in internal flow also develops over a certain length, known as the , before reaching a fully developed state
- In the thermal entry region, the temperature profile changes from a uniform profile at the inlet to a fully developed profile downstream
- The shape of the fully developed temperature profile depends on the boundary conditions (constant heat flux or constant wall temperature)
- The (Nu) is a dimensionless parameter that represents the ratio of convective to conductive heat transfer in internal flows
- Nu = ( × characteristic length) / thermal conductivity
- Higher Nusselt numbers indicate more effective convective heat transfer compared to conductive heat transfer
Friction Factor and Surface Roughness Effects
- The (f) is a dimensionless parameter that represents the resistance to fluid flow in pipes and ducts, and it depends on the Reynolds number (Re) and the relative roughness of the surface
- f = (pressure drop × diameter) / (density × velocity^2 × length)
- The friction factor is higher for rough surfaces compared to smooth surfaces, leading to increased pressure drop and pumping power requirements
- In , the friction factor is a function of the Reynolds number only (f = 64/Re for circular pipes)
- In turbulent flow, the friction factor depends on both the Reynolds number and the relative roughness (Moody diagram)
Entry Lengths for Internal Flows
Hydrodynamic Entry Length
- The hydrodynamic entry length (Lh) is the distance from the inlet of a pipe or duct to the point where the velocity profile becomes fully developed
- For laminar flow, the hydrodynamic entry length can be estimated using the equation: Lh/D ≈ 0.05 * Re, where D is the diameter of the pipe or duct
- Example: For a laminar flow with Re = 1000 in a pipe with D = 0.02 m, Lh ≈ 0.05 * 1000 * 0.02 = 1 m
- For turbulent flow, the hydrodynamic entry length is much shorter and can be estimated using the equation: Lh/D ≈ 10
- Example: For a turbulent flow in a pipe with D = 0.02 m, Lh ≈ 10 * 0.02 = 0.2 m
- The hydrodynamic entry length is important for determining the pressure drop and pumping power requirements in internal flows
Thermal Entry Length
- The thermal entry length (Lt) is the distance from the inlet of a pipe or duct to the point where the thermal boundary layer becomes fully developed
- The thermal entry length depends on the (Pr) and can be estimated using the equation: Lt/D ≈ 0.05 * Re * Pr, for laminar flow
- Example: For a laminar flow with Re = 1000 and Pr = 0.7 in a pipe with D = 0.02 m, Lt ≈ 0.05 * 1000 * 0.7 * 0.02 = 0.7 m
- In turbulent flow, the thermal entry length is shorter than the laminar case and can be estimated using the equation: Lt/D ≈ 10
- Example: For a turbulent flow in a pipe with D = 0.02 m, Lt ≈ 10 * 0.02 = 0.2 m
- The thermal entry length is important for determining the heat transfer characteristics and the effectiveness of heat exchangers
Heat Transfer Coefficients in Internal Flows
Laminar Flow Correlations
- The Nusselt number for fully developed laminar flow in a circular pipe with constant heat flux can be calculated using the : Nu = 1.86 * (Re * Pr * D/L)^(1/3) * (μ/μs)^0.14, where μ is the fluid viscosity at the bulk temperature and μs is the fluid viscosity at the surface temperature
- This correlation accounts for the variation of fluid properties with temperature and is valid for 0.48 ≤ Pr ≤ 16,700 and Re * Pr * D/L ≥ 10
- Example: For a laminar flow with Re = 1000, Pr = 0.7, D = 0.02 m, L = 1 m, and (μ/μs) = 0.8, Nu ≈ 1.86 * (1000 * 0.7 * 0.02/1)^(1/3) * 0.8^0.14 ≈ 4.64
- Other correlations for laminar flow in circular pipes include the Hausen correlation and the Leveque solution, which are applicable for different boundary conditions and flow regimes
Turbulent Flow Correlations
- For fully developed turbulent flow in a circular pipe, the can be used to calculate the Nusselt number: Nu = 0.023 * Re^(4/5) * Pr^n, where n = 0.4 for heating and n = 0.3 for cooling
- This correlation is valid for 0.6 ≤ Pr ≤ 160, Re ≥ 10,000, and L/D ≥ 10
- Example: For a turbulent flow with Re = 50,000, Pr = 0.7, and heating, Nu ≈ 0.023 * 50,000^(4/5) * 0.7^0.4 ≈ 153.5
- The is a more accurate alternative to the Dittus-Boelter correlation for turbulent flow in circular pipes, valid for 3,000 ≤ Re ≤ 5 × 10^6 and 0.5 ≤ Pr ≤ 2,000
- Nu = (f/8) * (Re - 1000) * Pr / (1 + 12.7 * (f/8)^(1/2) * (Pr^(2/3) - 1)), where f is the friction factor
- Example: For a turbulent flow with Re = 50,000, Pr = 0.7, and f = 0.018, Nu ≈ (0.018/8) * (50,000 - 1000) * 0.7 / (1 + 12.7 * (0.018/8)^(1/2) * (0.7^(2/3) - 1)) ≈ 148.8
- The is another accurate correlation for turbulent flow in circular pipes, valid for 3,000 ≤ Re ≤ 5 × 10^6 and 0.5 ≤ Pr ≤ 200
- Nu = (f/8) * Re * Pr / (1.07 + 12.7 * (f/8)^(1/2) * (Pr^(2/3) - 1))
- Example: For a turbulent flow with Re = 50,000, Pr = 0.7, and f = 0.018, Nu ≈ (0.018/8) * 50,000 * 0.7 / (1.07 + 12.7 * (0.018/8)^(1/2) * (0.7^(2/3) - 1)) ≈ 153.2
Convective Heat Transfer in Non-Circular Ducts
Hydraulic Diameter and Its Application
- Non-circular ducts, such as rectangular or triangular cross-sections, are commonly encountered in heat exchangers and other industrial applications (plate-fin heat exchangers, compact heat exchangers)
- The (Dh) is used to characterize the dimensions of non-circular ducts, defined as Dh = 4A/P, where A is the cross-sectional area and P is the wetted perimeter
- Example: For a with width a = 0.1 m and height b = 0.05 m, Dh = 4 * (0.1 * 0.05) / (2 * (0.1 + 0.05)) ≈ 0.0667 m
- The correlations for circular pipes can be used for non-circular ducts by replacing the diameter (D) with the hydraulic diameter (Dh)
- Example: The Dittus-Boelter correlation for turbulent flow in a rectangular duct: Nu = 0.023 * Re^(4/5) * Pr^n, where Re and Nu are based on the hydraulic diameter
Heat Transfer in Annuli
- Annuli, or concentric circular pipes, are another common configuration in heat exchangers (double-pipe heat exchangers, shell-and-tube heat exchangers)
- The hydraulic diameter for an annulus is defined as Dh = Do - Di, where Do is the outer diameter and Di is the inner diameter
- Example: For an annulus with Do = 0.05 m and Di = 0.03 m, Dh = 0.05 - 0.03 = 0.02 m
- Correlations for heat transfer in annuli can be derived from the correlations for circular pipes by using the hydraulic diameter and appropriate modifications to account for the geometry
- Example: The Dittus-Boelter correlation for turbulent flow in an annulus: Nu = 0.023 * Re^(4/5) * Pr^n * (Di/Do)^0.45, where Re and Nu are based on the hydraulic diameter
- The heat transfer in annuli is influenced by the ratio of the inner to outer diameter (Di/Do) and the direction of heat transfer (heating or cooling of the inner or outer wall)
Key Terms to Review (27)
Circular duct: A circular duct is a cylindrical passage used to convey fluids, typically air, in various engineering applications, particularly in HVAC systems. Its shape allows for efficient airflow, minimizing turbulence and pressure losses compared to other duct shapes. This efficiency is vital in systems designed for forced convection and internal flow, as it directly affects the heat transfer performance and energy consumption.
Continuity equation: The continuity equation is a mathematical statement that expresses the principle of mass conservation within a fluid flow system. It ensures that mass cannot be created or destroyed in a control volume, meaning the mass entering a system must equal the mass exiting it. This principle applies to various flow situations, including external forced convection, internal forced convection, and diffusion processes, highlighting the interconnectedness of mass transfer in different contexts.
Convective Heat Transfer Coefficient: The convective heat transfer coefficient is a measure of the heat transfer between a solid surface and a fluid flowing over it. This coefficient depends on the nature of the flow, the properties of the fluid, and the characteristics of the surface, making it crucial for understanding how heat is transferred in various situations involving convection.
Cooling Systems: Cooling systems are mechanisms designed to remove excess heat from a designated area, ensuring that temperatures remain within acceptable limits for operational efficiency and safety. These systems are crucial in various applications, including industrial processes, electronics cooling, and HVAC systems, as they facilitate heat transfer through forced convection, either externally or internally. Understanding the principles of forced convection in cooling systems is essential to optimize their performance and ensure effective heat management.
Dittus-Boelter Correlation: The Dittus-Boelter Correlation is an empirical relationship used to calculate the convective heat transfer coefficient for turbulent flow inside smooth tubes. This correlation is crucial for understanding heat transfer in various engineering applications, as it connects fluid dynamics and thermal characteristics of flow systems. Its relevance spans across boundary layer behavior, forced convection scenarios, and optimization in heat exchanger design, making it an essential tool for engineers and designers.
Friction Factor: The friction factor is a dimensionless quantity used to quantify the resistance to flow in a fluid system, particularly in internal flow situations. It is crucial for determining pressure drops and energy losses in forced convection systems, as it helps relate the velocity, density, and viscosity of the fluid to the geometry of the conduit. Understanding the friction factor allows for better designs and optimizations in systems where fluid flow is involved.
Gnielinski Correlation: The Gnielinski correlation is an empirical relationship used to calculate the convective heat transfer coefficient for fluid flow inside pipes, particularly under turbulent conditions. This correlation is crucial for determining heat transfer rates in internal flows, as it provides a means to relate the Nusselt number to the Reynolds number and the Prandtl number, which are essential for analyzing convection phenomena and boundary layer behavior.
Grashof Number: The Grashof number is a dimensionless quantity used in fluid mechanics to characterize the ratio of buoyant forces to viscous forces within a fluid. It plays a crucial role in understanding natural convection phenomena, indicating whether buoyancy-driven flow is significant compared to viscous effects. This number helps determine flow regimes and influences heat transfer rates in various fluid situations, especially where temperature differences lead to density variations.
Heat Exchangers: Heat exchangers are devices designed to efficiently transfer heat from one medium to another, often between liquids or gases, without mixing them. They play a crucial role in various applications, such as in heating, cooling, and energy recovery systems, facilitating the transfer of thermal energy through conduction and convection.
Hydraulic Diameter: The hydraulic diameter is a characteristic length used in fluid mechanics to define the effective cross-sectional area of a non-circular conduit or channel, crucial for analyzing fluid flow. It is calculated using the formula $$D_h = \frac{4A}{P}$$, where $$A$$ is the cross-sectional area and $$P$$ is the wetted perimeter. This term is essential in forced convection within internal flows, as it influences the flow characteristics, heat transfer rates, and pressure drop within ducts or pipes.
Hydrodynamic Entry Length: Hydrodynamic entry length refers to the distance a fluid travels within a duct or pipe before it fully develops a stable velocity profile and reaches a state of fully developed flow. This concept is crucial when analyzing forced convection in internal flow systems, as it affects the heat transfer characteristics and pressure drop along the flow path. Understanding entry length helps engineers design more efficient systems by optimizing flow conditions from the onset.
Laminar flow: Laminar flow is a type of fluid motion characterized by smooth, parallel layers of fluid that move in an orderly fashion without disruption between them. This type of flow occurs at low velocities and is typically seen in situations where fluid viscosity is high or flow area is small. The behavior of laminar flow is significant for understanding how fluids interact with surfaces, heat transfer, and mass transfer processes.
Navier-Stokes Equations: The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of fluid substances, taking into account viscosity and other forces acting on the fluid. These equations are fundamental in understanding fluid dynamics and play a crucial role in modeling various phenomena related to heat and mass transfer in both forced and natural convection processes, as well as in the study of mass transport and at microscale levels. They provide the mathematical framework for analyzing complex flow patterns, predicting behavior in different conditions, and facilitating computational fluid dynamics simulations.
Nusselt Number: The Nusselt number is a dimensionless quantity used in heat transfer that represents the ratio of convective to conductive heat transfer across a boundary. It helps to characterize the efficiency of convective heat transfer in fluid flows, making it essential for understanding processes involving both heat and mass transfer.
Overall Heat Transfer Coefficient: The overall heat transfer coefficient is a measure of a material's ability to conduct heat across its thickness, accounting for conduction, convection, and sometimes radiation. It combines the effects of the various resistances to heat flow, making it essential for analyzing systems like heat exchangers or internal flow setups. Understanding this coefficient helps predict how efficiently heat will transfer through a system, particularly in forced convection scenarios and during thermal analyses using the effectiveness-NTU method.
Petukhov Correlation: The Petukhov correlation is a widely used empirical relationship for predicting the Nusselt number in fully developed turbulent flow inside circular pipes. This correlation helps in estimating convective heat transfer coefficients, which are crucial for understanding heat transfer in internal flows, particularly in engineering applications involving fluid transport in pipes.
Prandtl Number: The Prandtl number is a dimensionless number that measures the relative thickness of the momentum boundary layer to the thermal boundary layer in a fluid. It helps characterize the heat transfer and fluid flow properties in convection processes, highlighting the relationship between momentum diffusivity (viscosity) and thermal diffusivity (heat conduction). Understanding the Prandtl number is crucial for analyzing various heat transfer scenarios, especially in both forced and natural convection.
Rectangular Duct: A rectangular duct is a type of conduit used for transporting fluids, particularly air, in heating, ventilation, and air conditioning (HVAC) systems. This shape allows for efficient airflow and optimal use of space in buildings, making it a popular choice for internal flow applications. The design of rectangular ducts can significantly influence the characteristics of forced convection, affecting parameters such as pressure drop, velocity distribution, and heat transfer rates.
Reynolds Number: Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations by comparing inertial forces to viscous forces. It is a critical factor in determining whether the flow is laminar or turbulent, influencing heat and mass transfer rates in various contexts.
Sieder-Tate Correlation: The Sieder-Tate correlation is an empirical relationship used to estimate the convective heat transfer coefficient in internal flow situations, especially in turbulent flow through ducts and pipes. This correlation helps in analyzing heat transfer rates by relating the Nusselt number to the Reynolds and Prandtl numbers, which are key factors in characterizing fluid flow and heat transfer behavior. It's particularly useful in engineering applications where accurate predictions of thermal performance are critical.
Surface Roughness: Surface roughness refers to the texture of a surface, characterized by the irregularities and deviations from a perfectly smooth plane. It plays a crucial role in determining how fluids interact with solid boundaries, influencing factors like drag, heat transfer rates, and overall performance in thermal and fluid systems. Additionally, surface roughness affects the absorption and emission of thermal radiation, which is vital for understanding energy exchange in real-world applications.
Thermal boundary layer: The thermal boundary layer is the region in a fluid where temperature changes from the value of the fluid away from a surface to the temperature of that surface. This layer is crucial in understanding heat transfer, as it influences convection and the effectiveness of heat exchange between a solid and a fluid. The characteristics of this layer can significantly affect the heat transfer coefficients and the overall thermal performance of systems.
Thermal Entry Length: Thermal entry length is the distance along a flow path within a duct or pipe where the fluid temperature adjusts to the wall temperature, resulting in a fully developed thermal profile. This length is crucial in internal forced convection as it determines how quickly the fluid achieves thermal equilibrium with the duct walls, influencing heat transfer rates and efficiency within the system.
Thermal Imaging: Thermal imaging is a technology that detects and visualizes the infrared radiation emitted by objects, allowing for the measurement of temperature differences and thermal patterns. This method is essential in analyzing heat transfer phenomena, especially in scenarios where temperature distribution is critical, such as in forced convection and solving inverse heat transfer problems. By translating thermal energy into visual representations, thermal imaging aids in identifying hotspots, inefficiencies, and potential issues in various systems.
Turbulent flow: Turbulent flow is a type of fluid motion characterized by chaotic changes in pressure and flow velocity. In this state, the fluid exhibits irregular fluctuations and eddies, making it quite different from laminar flow where the fluid moves in smooth layers. Turbulent flow plays a crucial role in heat transfer, mixing, and mass transfer processes, affecting the performance and efficiency of many engineering systems.
Velocity Profile: A velocity profile is a graphical representation that shows how fluid velocity varies across different positions within a flow field. Understanding the velocity profile is crucial because it helps to characterize flow behavior, predict pressure drops, and determine heat transfer rates, especially in boundary layers and forced convection scenarios.
Wind Tunnel Testing: Wind tunnel testing is a method used to study the aerodynamic properties of objects by simulating the effects of wind on them in a controlled environment. This technique is crucial for analyzing flow patterns, drag forces, and heat transfer characteristics, providing valuable data for improving designs and performance in various engineering applications.