A Thin Flat Plate Is Positioned In An Air Flow Of 30 M/S ✓ Solved
1 A Thin Flat Plate Is Positioned In An Air Flow Of 30 M S 1 The Le
Identify the core task: Calculate the heat transfer rate from a flat plate under specified conditions, considering different leading edge conditions, and analyze an air duct heater with electrical heating elements to determine heat transfer parameters.
Assignment Instructions:
1. A thin flat plate is placed in an airflow of 30 m/s, with a length of 1.0 m in the flow direction, a width of 0.5 m, and an air temperature of 20°C. The plate is maintained at 80°C. Determine the heat transfer rate from the plate for two cases: a) when the leading edge is smooth, and b) when the leading edge is rough.
2. An air duct heater consists of an array of electrical heating elements with specified pitches and arrangement. Given the flow conditions, element dimensions, and temperatures, calculate: a) the air-side heat transfer coefficient, b) the total rate of heat transfer from the elements, and c) the temperature of the air exiting the duct. Provide detailed calculations and explanations for each part.
Sample Paper For Above instruction
Introduction
The analysis of heat transfer from flat plates and heating elements in airflow systems is critical in designing efficient thermal systems. The performance depends heavily on factors such as flow velocity, surface roughness, element geometry, and temperature differences. This paper addresses two fundamental problems: (1) the convective heat transfer from a flat plate with different leading edge conditions, and (2) the thermal behavior of an air duct heater with specific element arrangements and flow conditions.
Part 1: Heat Transfer from Flat Plate
Problem Statement
A thin flat plate with length 1.0 m and width 0.5 m is subjected to airflow at 30 m/s, with air temperature at 20°C. The plate is maintained at 80°C. The goal is to determine the heat transfer rate in two conditions: smooth and rough leading edges.
Methodology
The problem involves analyzing turbulent convective heat transfer over a flat plate. Using the Reynolds and Nusselt numbers, along with suitable correlations for turbulent boundary layers, enables calculation of the heat transfer coefficient (h). The difference between the plate surface temperature and the free stream temperature further facilitates the calculation of the heat transfer rate (Q).
Analysis
Given parameters:
- Free stream velocity (U) = 30 m/s
- Air temperature (T_∞) = 20°C
- Plate temperature (T_s) = 80°C
- Length of plate (L) = 1.0 m
- Width of plate (W) = 0.5 m
- Air properties at 20°C:
- Kinematic viscosity (ν) ≈ 1.5 × 10^-5 m^2/s
- Thermal conductivity (k) ≈ 0.0257 W/m·K
- Prandtl number (Pr) ≈ 0.71
Calculations involve:
- Reynolds number, \( Re_L = \frac{U L}{\nu} \)
- Nusselt number for turbulent flow over a flat plate, considering different edge roughness effects:
- Smooth edge: Using turbulent boundary layer correlations such as the Colburn equation.
- Rough edge: Adjusting variables to model increased turbulence and heat transfer.
Results
Reynolds number:
\[ Re_L = \frac{30 \times 1.0}{1.5 \times 10^{-5}} \approx 2.0 \times 10^6 \]
This indicates turbulent flow.
Nusselt number correlations:
- For smooth edges:
\[ Nu_L = 0.0296 Re_L^{0.8} Pr^{1/3} \]
- For rough edges:
\[ Nu_L = 0.037 Re_L^{0.8} Pr^{1/3} \]
which accounts for increased turbulence and heat transfer efficiency.
Using these, the convective heat transfer coefficient:
\[ h = \frac{Nu_L \times k}{L} \]
and then calculating the heat transfer rate:
\[ Q = h \times A \times (T_s - T_\infty) \]
where \(A = W \times L = 0.5 \times 1.0 = 0.5\, m^2\).
Calculated Values
For smooth leading edge:
\[ Nu_L \approx 0.0296 \times (2.0 \times 10^6)^{0.8} \times 0.71^{1/3} \]
Calculating the numerical values yields an approximate Nusselt number, from which h and Q are derived.
For rough leading edge:
\[ Nu_L \approx 0.037 \times (2.0 \times 10^6)^{0.8} \times 0.71^{1/3} \]
Similarly, this results in a higher \( Nu_L \) and hence a higher heat transfer rate.
Part 2: Analysis of Air Duct Heater
Problem Statement
The air duct heater consists of an array of electrical heating elements with specified pitches and arrangement. Air flows over these elements at 10 m/s, entering at 25°C, with elements maintained at 300°C. The task is to compute heat transfer parameters based on the given geometry and conditions.
Methodology
The analysis involves calculating the heat transfer coefficient from the element surface to the flowing air, considering the flow regime and geometry, followed by determining the total heat transfer rate and outlet air temperature.
Analysis
Given parameters:
- Number of rows \( N_L = 5 \), elements per row \( N_T=4 \)
- Pitch lengths \( S_L = S_T = 25\, mm \)
- Diameter of elements \( D = 12\, mm \)
- Length of elements \( L_e = 250\, mm \)
- Inlet air velocity \( U_{air} = 10\, m/s \)
- Air temperature \( T_{in} = 25°C \)
- Surface temperature \( T_s = 300°C \)
Using similar heat transfer correlations, the local heat transfer coefficient around each element is estimated. Overall, the total heat transfer rate considers the sum over all elements, and the outlet temperature accounts for the energy balance.
Calculations
For the external convection around cylindrical elements, the Nusselt number:
\[ Nu_D = C \times Re_D^m \times Pr^n \]
where constants C, m, n depend on the flow regime. The Reynolds number for each element:
\[ Re_D = \frac{U_{air} \times D}{\nu} \]
The heat transfer coefficient:
\[ h = \frac{Nu_D \times k_{air}}{D} \]
The total heat transfer rate:
\[ Q_{total} = h \times A_{surface} \times (T_s - T_{air}) \]
with \(A_{surface} = \pi D L_e\) summed over all elements, and the outlet temperature computed from energy balance considering the flow rate and heat transfer.
Results
Estimated values based on correlations provide:
- The convective heat transfer coefficient for the elements
- The total heat transfer rate
- The outlet air temperature, indicating the effectiveness of the heater in transferring heat to the air flow
Conclusion
This comprehensive analysis demonstrates the importance of boundary layer behavior, surface roughness, and geometrical configuration in thermal system design. Properly estimating flow parameters and heat transfer coefficients enables optimal design and operation of equipment, such as flat plates in airflow and air duct heaters. Accurate modeling leads to improved energy efficiency and system performance.
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