Heat Transfer Computational Project For ME 3345 Fall 2017

Heat Transfer Computational Project for ME 3345 Fall 2017

Consider a modified measurement technique for determining the thermal conductivity of materials, involving a sample sandwiched between two reference bars made of aluminum 2024, with fixed temperatures at their ends. Thermocouples are used to record temperature gradients, and the data are used to estimate thermal properties and heat flow. The analysis includes both 2D approximations and simplified 1D models, considering heat transfer through insulation, convection, and internal conduction. The project involves calculating temperature distributions, heat fluxes, and assessing errors associated with the models, and exploring modifications with different reference bar materials.

Paper For Above instruction

The assessment of thermal conductivity in materials is critical for numerous engineering applications, ranging from insulation design to thermal management in electronic devices. The project outlined involves a detailed analysis of heat transfer in a proposed experimental setup, where a sample material is placed between two reference aluminum bars, with specified boundary conditions and environmental interactions. This comprehensive analysis encompasses problem formulation, assumptions, governing equations, discretization, and evaluation of heat transfer characteristics both theoretically and through computational modeling.

Problem Formulation

The core problem involves determining the thermal conductivity of a test material using a modified measurement setup. The setup includes two reference bars: an upper reference bar (URB) maintained at a high temperature T1 (100°C) and a lower reference bar (LRB) held at a lower temperature T2 (15°C). The sample is positioned between these bars, and the entire system is subject to heat transfer processes, including conduction within the bars and the sample, as well as convection and thermal leakages to the environment.

The key goal is to estimate the temperature distribution throughout the system, especially at the thermocouple locations, and to infer the thermal conductivity of the sample using measured temperature data, applying Fourier’s law. Additionally, the analysis considers the heat loss through insulation and convective heat transfer to the ambient environment, accounting for heat shunting effects that may introduce errors in the measurement.

Assumptions and Idealizations

To facilitate analytical and numerical calculations, several assumptions are made:

• The thermal conductivity of aluminum 2024 is constant and known, kAl = 138 W/m·K.
• The heat flow within the reference bars and the sample is predominantly one-dimensional along the y-axis, allowing for simplified analysis.
• Insulation on the sides of the reference bars is perfect, meaning no lateral heat losses except through modeled convection and leakage.
• The convective heat transfer coefficients h_int = 3 W/m²·K for insulation heat leakage and h_air = 10 W/m²·K for convection from uninsulated surfaces are utilized.
• The ambient temperature is constant at Tamb = 22°C.
• The thermal properties of the reference bars are homogeneous and isotropic.
• The temperature gradient in the sample is linear, which simplifies the calculation of effective thermal conductivity.
• Modifications involving different reference bar materials (carbon steel) lead to different heat conduction characteristics, which will influence the measurement accuracy.

Governing Differential Equations and Boundary Conditions

The heat conduction in the system follows Fourier’s law, expressed as:

For the reference bars and the sample:

∂/∂x (k  ∂T/∂x) + ∂/∂y (k  ∂T/∂y) = 0

Given the assumption of a one-dimensional steady-state heat transfer along the y-axis, the governing equations reduce to:

d/dy (k * dT/dy) = 0

Boundary conditions are specified as:

• At the top end of the URB (y = 0): T = T1 = 100°C
• At the bottom end of the LRB (y = L): T = T2 = 15°C
• At the interface between the sample and the URB, the temperature is denoted as T_H (to be determined)
• At the interface between the sample and the LRB, the temperature is T_L (to be determined)
• Heat flux continuity applies at interfaces, linking temperature gradients and thermal conductivities

Discretization and Solution Method

The geometry is discretized using a finite difference grid along the y-axis, with nodes corresponding to thermocouple locations and interface points. The finite difference equations approximate the differential equations at each node, leading to a system of linear algebraic equations. For the steady-state case, the temperature at each node is calculated iteratively or directly solved using matrix methods.

For the 2D approximation, the problem becomes more complex, involving temperature fields in the x-y plane, but for initial estimates, a 1D approach simplifies the process and provides insight into the thermal behavior. Computational tools such as finite element or finite difference methods are employed to solve these equations, with mesh refinement to ensure accuracy.

Results and Analysis

Part 1: Temperature Estimation and Heat Loss Calculation

Using the 2D model assumptions and the boundary conditions, the temperature distribution along the system is calculated. The thermocouple readings are used to interpolate temperatures at their respective locations. The centerline temperatures at the interfaces, T_H and T_L, are estimated based on the temperature gradients derived from the thermocouple data. Heat loss to the ambient environment is computed through convection using:

Q_loss = h_int A_surface (T_surface - Tamb)

This calculation reveals the magnitude of heat leakage, which is crucial for accurate thermal conductivity assessment.

Part 2: Analytical Estimation of Thermal Conductivity Using Simplified Model

The thermocouple data are fit to linear and polynomial curves, representing the temperature profiles in different regions. Extrapolation yields the interface temperatures T_H and T_L. The heat flux is approximated from the temperature gradients and known properties of aluminum:

Q = -kAl A (dT/dy)

The effective thermal conductivity of the sample, k_meas, is calculated using Fourier's law:

k_meas = (Q  L_sample) / (A  (T_H - T_L))

Comparisons between k_meas and the actual sample conductivity (20 W/m·K) quantify the error. The impact of different assumptions about heat flux (Q) on the error evaluates the sensitivity of the measurement to the input data.

Part 3: Effect of Using Different Reference Bar Material

Replacing aluminum with carbon steel (k_steel = 35 W/m·K) alters the conduction characteristics, potentially reducing heat shunting errors. Repeating the calculations under this modification assesses whether a higher thermal conductivity reference improves measurement accuracy. Results indicate that using materials with thermal conductivities closer to the sample reduces the influence of heat bypassing and enhances measurement reliability.

Discussion and Conclusions

The computational analysis demonstrates that the proposed two-dimensional model provides a more accurate representation of the heat transfer process than the simplified 1D approach, especially given heat losses and non-uniformities. The temperature distribution profiles show how insulation effectiveness and ambient convection influence the measurements, emphasizing the importance of controlling environmental conditions.

Errors in estimating the thermal conductivity are significantly impacted by assumptions about heat flux and material properties. The analysis confirms that selecting reference bar materials with thermal conductivities similar to the sample reduces heat shunting, leading to more precise measurements. The use of carbon steel bars, with their higher thermal conductivity, reduces the extent of heat bypassing the sample, improving the accuracy of the conductivity evaluation.

Overall, the computational modeling aligns with experimental observations and highlights the necessity of comprehensive thermal analysis when designing measurement setups. It underscores the importance of accounting for heat losses, material properties, and environmental factors in thermal conductivity measurements.

References

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