Thermal-Fluid Systems
(Level 5)
Previous levels: motion, forces, rotation, vibration. All about mechanical behavior. Level 5 asks different questions. How does heat move? How do fluids flow? How does energy get transported through systems? Why does temperature matter for material performance, stress, and failure?
This is transport physics. Energy, mass, and momentum moving through space and time. Everything couples together—heat affects flow, flow affects heat, pressure affects both. Unlike isolated point masses, these are continuous systems. You can't ignore spatial variation. You can't always write closed-form solutions. Welcome to why CFD and FEA exist.
Thermodynamic Systems and Energy Balance
Draw a boundary around whatever you're analyzing. That's your system. Everything outside is the surroundings. Energy crosses the boundary. Work crosses the boundary. Sometimes mass crosses too (open system). The fundamental rule: energy is conserved.
Sounds simple. It is simple. But it's also everything. Heat engine? Energy balance. Heat exchanger? Energy balance. Battery charging? Energy balance. Building heating up in summer? Energy balance. The equation structure stays the same. The complexity is in identifying what crosses the boundary and how.
Q_in − Q_out + W_in − W_out = ΔU (change in internal energy)
Example: Sealed piston-cylinder. Heat added, work done by compression, internal energy changes.
Energy balance includes enthalpy carried by flowing mass.
Example: Turbine. Hot gas flows in with energy, does work, cooler gas flows out. Steady-state: energy flow in = energy flow out + work extracted.
This is the first law of thermodynamics in action. It doesn't tell you which direction energy flows (that's second law). It just says energy is accounted for. Master energy balances and you can analyze any thermal system, from refrigerators to power plants.
Heat Transfer
Heat flows from hot to cold. Always. Three mechanisms make it happen:
Heat moves through solid material. Molecules vibrate, energy transfers to neighbors. Touch a hot metal surface—conduction is why you burn your hand.
k is thermal conductivity. High k (copper, aluminum) → heat spreads fast. Low k (foam, fiberglass) → insulation. Temperature gradient dT/dx drives the flow. Steeper gradient → more heat transfer.
Heat transfers between a surface and a moving fluid (air, water, oil). Fluid carries energy away. Fan blowing on your skin? Convection.
h is convection coefficient (depends on flow speed, fluid properties, geometry). T_s is surface temperature, T_∞ is fluid temperature far from surface. Bigger temperature difference or higher h → more heat transfer. Forced convection (pump/fan) gives higher h than natural convection (buoyancy-driven flow).
Heat transfers as electromagnetic waves. No medium needed. Doesn't require contact or fluid flow. Sun heating Earth? Radiation. Glowing metal in a furnace? Radiation.
ε is emissivity (surface property, 0 to 1). σ is Stefan-Boltzmann constant. T⁴ dependence: high temperatures → radiation dominates. At room temperature, radiation is small compared to conduction/convection. At 500°C and above, radiation becomes significant.
Most real systems use all three modes at once. Engine block: conduction through metal, convection to coolant, radiation to surroundings. Electronics cooling: conduction from chip to heat sink, convection to air, sometimes radiation to enclosure. Analyzing heat transfer means identifying which modes matter and sizing them correctly.
Fluid Flow and Momentum Transport
Fluids flow when pressure differences exist or when body forces (gravity) act on them. Unlike solids, fluids deform continuously under shear stress. Flow is described by velocity fields, pressure fields, density distributions—all varying in space and time.
Conservation of mass (continuity equation):
Incompressible flow (constant density): A₁v₁ = A₂v₂. Pipe narrows? Flow speeds up. Flow rate is constant, area decreases, velocity increases. This is why garden hose nozzles work—restrict the area, velocity shoots up.
Diameter halves → area drops by 4× → velocity increases 4× to maintain same mass flow rate.
Momentum conservation leads to more complex equations. Navier-Stokes equations govern viscous fluid flow: they're partial differential equations combining pressure gradients, viscous stresses, and inertia. For simple cases (steady, laminar, low Reynolds number), you get tractable solutions. For most real flows (turbulent, complex geometry, unsteady), you need CFD.
Bernoulli's equation (simplified momentum balance for inviscid flow along a streamline):
Pressure + kinetic energy per volume + potential energy per volume = constant. Fast flow → low pressure (airplane wing lift). Slow flow → high pressure. Elevation increases → pressure decreases. Trade-offs between pressure, velocity, and elevation govern everything from pumps to airfoils to water towers.
Transport Phenomena
Heat transfer, fluid flow, mass diffusion—all governed by similar math. Transport phenomena unifies them. Each has a flux (flow per unit area), a driving gradient (temperature, velocity, concentration), and a transport property (conductivity, viscosity, diffusivity).
Heat flux: q″ = −k ∇T (Fourier's law)
Momentum flux (shear stress): τ = μ dv/dy (Newton's law of viscosity)
Mass flux: J = −D ∇C (Fick's law)
Same pattern: flux proportional to gradient. This isn't coincidence—it's fundamental physics of diffusion-like processes. Gradients drive flows. Steeper gradient → stronger flux. Transport property determines how efficiently the gradient produces flux.
Why This Matters Later
Techniques for solving heat conduction problems often work for mass diffusion or momentum transport. Dimensionless numbers (Prandtl, Schmidt, Lewis numbers) relate these transport processes. Coupled problems—like combustion (heat + mass + flow) or drying (heat + mass)—are analyzed using unified transport theory.
Why Numerical Methods Matter
Simple geometry, steady-state, constant properties, linear behavior? You might get an analytical solution. Textbook problems live here. Real engineering doesn't.
Complex geometry (actual ducts, heat sinks, turbine blades). Time-varying conditions (startup, shutdown, transient response). Nonlinear material properties (temperature-dependent conductivity, non-Newtonian fluids). Turbulence (almost all high-speed flows). Coupled physics (thermal expansion affecting flow, flow affecting heat transfer). Exact solutions? Forget it.
This is why CFD (Computational Fluid Dynamics) and FEA (Finite Element Analysis) exist. Discretize the domain into small cells or elements. Apply conservation laws locally. Solve the resulting (huge) system of algebraic equations numerically. Iterate until convergence.
1. Mesh the geometry (break it into thousands or millions of cells)
2. Apply governing equations at each cell
3. Impose boundary conditions
4. Solve iteratively (linear solver or nonlinear iteration)
5. Check convergence and validate results
Software does the computation. Your job: set up the problem correctly, understand what the physics should look like, recognize when results are nonsense. Bad mesh? Wrong boundary conditions? Unconverged solution? Garbage output. Physics understanding is what separates an engineer from someone who just clicks buttons. Numerical methods let you solve hard problems, but only if you know what you're solving for.
Worked Example: Energy Balance on a Thermal System
System receives 500 W of heat input. Loses 200 W through the walls. No work done by or on the system. Find the rate of change of stored energy and explain what's happening.
Substitute: 500 W in, 200 W out
Rate of energy storage: 300 W
Positive dE/dt means energy is accumulating. System is heating up. Internal energy (and temperature) increasing over time. If this continues, system eventually reaches equilibrium when heat loss matches heat input, or something fails (overheats, melts, catches fire).
This is a lumped system analysis—treats the whole system as uniform temperature. No spatial details needed, just energy flows across boundaries. Good approximation when internal temperature gradients are small compared to surface-to-surroundings temperature difference (low Biot number). Otherwise you need spatial modeling.
Lesson: Energy balance is always the starting point. Identify what crosses the boundary, apply conservation, solve for unknowns. Most thermal analysis starts here, then adds complexity (spatial variation, time dependence, material properties, boundary conditions) as needed.
Physics Fundamentals Complete
Five levels done. You now have the physics foundation for mechanical engineering:
- Motion and vectors (Level 1): Kinematics, describing how things move
- Forces and energy (Level 2): Dynamics, why motion changes
- Rotation and systems (Level 3): Torque, inertia, multi-body analysis
- Vibrations and waves (Level 4): Oscillations, resonance, time-dependent behavior
- Thermal-fluid systems (Level 5): Heat, flow, transport, energy conversion
This covers the physics you need to understand statics, dynamics, mechanics of materials, fluid mechanics, heat transfer, thermodynamics, and vibrations. Not the full courses—just the essential physics that makes those courses make sense.
Engineering Physics Complete
You've completed all five levels of Engineering Physics. Review previous levels or explore other fundamentals.
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