Me 341 Heat Transfer For Engineering Exam 3 Page 2 Of 91
Me 341 Heat Transfer For Engineering Exam 3page 2 Of 91 You Are Taske
Calculate the boundary layer thicknesses, friction and heat transfer coefficients, heat transfer rate, and produce qualitative profiles for a flat plate in a forced convection scenario. Additionally, analyze heat transfer through a glass building side and a concentric tube heat exchanger involving oil and steam, all under steady-state conditions with specified parameters and properties.
Paper For Above instruction
Introduction
Understanding heat transfer phenomena in engineering systems is fundamental for designing efficient thermal management solutions. The behavior of convective boundary layers over surfaces, heat transfer through building materials, and heat exchange in tube systems are core areas within heat transfer engineering. This paper addresses multiple aspects of heat transfer, including boundary layer development on a flat plate, thermal analysis of a building facade, and the performance evaluation of a concentric tube heat exchanger. The analyses incorporate both qualitative and quantitative methods, employing principles of fluid mechanics, conduction, and convection to evaluate thermal characteristics under given conditions.
Boundary Layer Characteristics over a Flat Plate
A flat plate aligned in a uniform flow field serves as a classic example for studying forced convection heat transfer. The key parameters are the free-stream velocity, surface temperature, and fluid properties. At x = 0.3 meters, the viscous and thermal boundary layers develop along the plate length. The Reynolds number (Re), based on the local distance x, determines the laminar or turbulent nature of the flow and the transition point is observed at a critical Re (Re_critical).
The boundary layer thicknesses are derived from classical boundary layer theory equations. For laminar flow, the hydrodynamic boundary layer thickness \(\delta\) can be calculated as:
\[
\delta = 5 \frac{x}{\sqrt{Re_x}}
\]
where \(Re_x = \frac{\rho V x}{\mu}\). Assuming the properties at film temperature, the boundary layer at \(x=0.3\,\text{m}\) can be computed with known fluid properties like density \(\rho\), dynamic viscosity \(\mu\), and flow velocity \(V=3\,\text{m/s}\).
The thermal boundary layer \(\delta_t\) generally follows the relation:
\[
\delta_t = \frac{\delta}{\text{Pr}^{1/3}}
\]
with Prandtl number \(Pr\), which indicates the ratio of momentum to thermal diffusivity. Parameters are evaluated at the transition point corresponding to \(Re_{critical} = 5 \times 10^5\).
The transition from laminar to turbulent flow impacts the boundary layer thicknesses significantly. The boundary layer becomes thinner as the flow becomes turbulent due to increased mixing, affecting heat transfer rates.
Friction and Heat Transfer Coefficients
The average skin friction coefficient \(C_f\) in laminar flow is given by:
\[
C_f = \frac{1.328}{\sqrt{Re_x}}
\]
and in turbulent flow by empirical relations such as:
\[
C_f = 0.027 Re_x^{-1/7}
\]
The average convection heat transfer coefficient \(h\) is derived from:
\[
Nu_x = \frac{h x}{k}
\]
where \(Nu_x\) is the Nusselt number, obtained from correlations relevant to the flow regime:
- Laminar: \(Nu_x = 0.332 Re_x^{1/2} Pr^{1/3}\)
- Turbulent: \(Nu_x = 0.0296 Re_x^{0.8} Pr^{1/3}\)
Using these, average values of \(h\) over the length can be computed for both sections—before and after transition.
Heat Transfer Rate
The heat transfer rate per unit width can be determined using:
\[
Q'' = h \times (T_s - T_\infty)
\]
over the respective sections, considering the variations in \(h\) based on flow regimes. The total rate involves integrating the local heat fluxes along the plate length up to 0.3 m and the transition point. The convection heat transfer calculation reflects the significant increase in heat transfer coefficient upon transition to turbulence.
Qualitative Profiles of Friction and Heat Transfer Coefficients
Graphical representation depicts the friction coefficient \(C_f\) and the Nusselt number \(Nu_x\) along the plate. Initially, both parameters increase gradually in the laminar region, then sharply grow after the transition to turbulence, highlighting the enhanced momentum and heat transfer efficiencies in turbulent flow.
Heat Transfer through Glass Building Side
A transparent glass wall exposed to external cold air experiences heat loss primarily via convection and conduction (negligible conduction within the glass). Since the interior air temperature remains stable at 20°C, the exterior surface temperature stabilizes, influenced by external convective cooling at -15°C with a velocity of 15 m/s.
The analysis involves calculating the convective heat transfer coefficient \(h_{ext}\) using correlations such as:
\[
Nu = 0.3 + (0.62 Re^{1/2} Pr^{1/3}) / [1 + (0.4/Pr)^{2/3}]^{1/4} \times [1 + (Re/282000)^{5/8}]^{4/5}
\]
where Reynolds number \(Re = \frac{\rho V D}{\mu}\), and properties evaluated at the film temperature.
The overall heat loss per unit area is:
\[
Q = h_{ext} \times (T_{glass\_surface} - T_{outside})
\]
Since the interior temperature is maintained, the external convection dominates the heat transfer coefficients, which can be approximated accordingly, and the heat loss estimated.
Concentric Tube Heat Exchanger Performance
The heat exchanger analysis considers a thin-walled copper tube with oil flowing internally while steam condenses externally in the annulus, exchanging thermal energy. Given the inlet oil temperature, flow velocity, and the heat transfer coefficient for external condensation, the heat transfer per unit length is calculated by:
\[
Q' = U A \Delta T_{lm}
\]
where \(A = \pi D_{inner} \times 1\,\text{m}\), and \(\Delta T_{lm}\) is the Log Mean Temperature Difference between the oil inlet and steam condensate temperature, which can be approximated assuming steady-state conditions and similar properties at the respective temperatures.
The internal heat transfer coefficient of oil depends on its properties, which vary with temperature, and the convective heat transfer coefficient of steam condensation is assumed constant at 11,300 W/m²K. The overall heat transfer rate evaluates the effectiveness of the heat exchanger for specified flow rates and temperatures.
Conclusion
The comprehensive analysis illustrates the complex interplay of flow regimes, material properties, and boundary layer development affecting heat transfer in engineering systems. Precise calculations of boundary layer thicknesses, heat transfer coefficients, and heat exchange rates are critical for designing efficient thermal systems, whether for industrial heat exchangers or building insulation. Recognizing the transition points and accurately modeling the flow regime changes enable engineers to optimize performance and minimize energy consumption.
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