Heat Transfer, Thermomechanics and Coupled Fields
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Heat Transfer, Thermomechanics and Coupled Fields.
§6.1 Weak form of steady conduction
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `q=-k A dT/dx`. **Field study:** Calculate steady heat loss through a simple wall.
§6.2 Convection and thermal resistance
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `R=1/(hA)+L/(kA)`. **Field study:** Model a cooled electronic module through a conduction pad.
§6.3 Transient capacity and timestep selection
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `C ẋT + K T = Q`. **Field study:** Estimate an enclosure warm-up time constant.
§6.4 Thermomechanical sequential coupling
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `ε_th=α(T-T_ref)`. **Field study:** Estimate expansion of a heated mounting rail.
§6.5 Energy balances and thermal validation
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `Q_in-Q_out=dU/dt`. **Field study:** Compare modeled steady conduction against measured heater power.
Step-by-Step Finite Element Analysis and Computational Mechanics Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **one-dimensional heat conduction**, based on `Q=k*A*DeltaT/L`. Illustrative inputs are **Thermal conductivity=45 W/(mK), Cross section=0.04 m², Temperature difference=40 K, Thickness=0.02 m**. Converting the displayed units into SI inside the calculation gives **3600 W**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **one-dimensional heat conduction**, based on `Q=k*A*DeltaT/L`. Illustrative inputs are **Thermal conductivity=45 W/(mK), Cross section=0.04 m², Temperature difference=40 K, Thickness=0.02 m**. Converting the displayed units into SI inside the calculation gives **3600 W**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **one-dimensional heat conduction**, based on `Q=k*A*DeltaT/L`. Illustrative inputs are **Thermal conductivity=45 W/(mK), Cross section=0.04 m², Temperature difference=40 K, Thickness=0.02 m**. Converting the displayed units into SI inside the calculation gives **3600 W**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **conduction plus convection resistance**, based on `R=L/(kA)+1/(hA)`. Illustrative inputs are **Solid thickness=0.015 m, Conductivity=12 W/(mK), Film coefficient=25 W/(m²K), Area=0.15 m²**. Converting the displayed units into SI inside the calculation gives **0.275 K/W**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **conduction plus convection resistance**, based on `R=L/(kA)+1/(hA)`. Illustrative inputs are **Solid thickness=0.015 m, Conductivity=12 W/(mK), Film coefficient=25 W/(m²K), Area=0.15 m²**. Converting the displayed units into SI inside the calculation gives **0.275 K/W**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.