Thermal And Fluids

Reference notes on heat transfer, thermodynamics, and fluid mechanics.

Heat Transfer

Heat transfer describes thermal energy transport by conduction, convection, and radiation.

The transient heat equation for a homogeneous solid is:

$$\frac{\partial T}{\partial t}=\alpha {\mathrm{\nabla }}^{2}T+\frac{\dot{q}}{\rho c_p}$$

where $\alpha =k/(\rho c_p)$ is thermal diffusivity and $\dot{q}$ is volumetric heat generation.

Fourier conduction:

$$q_x=-kA\frac{dT}{dx}$$

Newton cooling:

$$q=hA(T_s-T_{\mathrm{\infty }})$$

Radiation to large surroundings:

$$q=ϵ\sigma A(T_s^4-T_{\text{sur}}^4)$$

Thermal resistance gives:

$$q=\frac{\mathrm{\Delta }T}{R_{\text{th}}}$$

For a plane wall:

$$R_{\text{cond}}=\frac{L}{kA}$$

For convection:

$$R_{\text{conv}}=\frac{1}{hA}$$

Lumped capacitance assumes spatially uniform body temperature. The Biot number checks whether internal gradients are likely important:

$$Bi=\frac{h{L}_c}{k}$$

A common lumped transient response is:

$$\frac{T(t)-T_{\mathrm{\infty }}}{T_i-T_{\mathrm{\infty }}}=\exp (-\frac{hA}{\rho V{c}_p}t)$$

Important dimensionless groups:

Group	Meaning
Biot	internal conduction resistance versus surface convection resistance
Fourier	nondimensional diffusion time
Nusselt	convection relative to conduction
Prandtl	momentum diffusivity versus thermal diffusivity

Thermal models depend strongly on geometry, material properties, boundary conditions, flow regime, characteristic length, and whether the response is steady or transient.

Extended surfaces or fins increase heat transfer area. Their usefulness depends on whether the added area remains thermally connected to the base. Fin efficiency compares actual fin heat transfer to an ideal fin at uniform base temperature.

For heat exchangers, the log-mean temperature difference method uses:

$$q=UA\mathrm{\Delta }{T}_{\text{lm}}$$

where $U$ is overall heat-transfer coefficient and $\mathrm{\Delta }{T}_{\text{lm}}$ accounts for the changing hot-cold temperature difference along the exchanger.

Convection correlations usually have the form:

$$Nu=f(Re,Pr,\text{geometry},\text{boundary condition})$$

Using a correlation outside its Reynolds-number range, geometry, surface condition, or thermal boundary condition can produce precise-looking but invalid heat-transfer estimates.

Thermodynamics

Thermodynamics tracks energy, work, heat, state, and property relationships.

For a closed system, the first law can be written:

$$\mathrm{\Delta }E=Q-W$$

where $Q$ is heat added to the system and $W$ is work done by the system.

For many engineering control volumes at steady state:

$$\dot{Q}-\dot{W}+\sum \dot{m}{(h+\frac{V^2}{2}+gz)}_{\text{in}}=\sum \dot{m}{(h+\frac{V^2}{2}+gz)}_{\text{out}}$$

State properties include pressure, temperature, volume, internal energy, enthalpy, entropy, and density. Processes may be idealized as isothermal, isentropic, isobaric, isochoric, adiabatic, or polytropic.

For an ideal gas:

$$pV=mRT$$

Enthalpy is:

$$h=u+pv$$

It is especially useful in open systems because flow work is included in the property.

Thermodynamics provides the bookkeeping for energy conversion. Heat transfer and fluid mechanics often provide the rate laws that determine how fast those changes occur.

The second law introduces entropy and limits on energy conversion. For a heat engine, thermal efficiency is:

$$\eta =\frac{W_{\text{out}}}{Q_{\text{in}}}$$

Thermodynamic models depend on property data, phase, ideal-gas assumptions, steady-flow assumptions, heat loss, irreversibility, and whether kinetic or potential energy terms are negligible.

For a reversible heat engine operating between two reservoirs, the Carnot efficiency is:

$${\eta }_{\text{Carnot}}=1-\frac{T_L}{T_H}$$

Real cycles fall below this limit because of irreversibility, finite temperature differences, friction, pressure losses, heat leakage, and non-ideal component behavior.

Common thermodynamic devices include nozzles, diffusers, turbines, compressors, pumps, throttling valves, heat exchangers, engines, refrigerators, and heat pumps. Each device has a characteristic energy balance and a set of loss mechanisms that determine performance.

Fluid Mechanics

Fluid mechanics relates pressure, velocity, density, viscosity, geometry, and flow rate.

For incompressible steady flow:

$$Q=AV$$

Ideal Bernoulli flow along a streamline:

$$\frac{p}{\rho g}+\frac{V^2}{2g}+z=\text{constant}$$

Pipe losses are often represented as:

$$h_f=f\frac{L}{D}\frac{V^2}{2g}$$

The Reynolds number compares inertial and viscous effects:

$$Re=\frac{\rho VL}{\mu }$$

Boundary-layer behavior governs convection, drag, skin friction, and separation. Laminar and turbulent regimes require different assumptions.

The incompressible continuity equation is:

$$\mathrm{\nabla }\cdot u=0$$

The Navier-Stokes momentum equation for a Newtonian incompressible fluid can be written:

$$\rho (\frac{\partial u}{\partial t}+u\cdot \mathrm{\nabla }u)=-\mathrm{\nabla }p+\mu {\mathrm{\nabla }}^{2}u+\rho g$$

Most practical fluid calculations are simplifications, correlations, or numerical approximations of mass and momentum conservation under specific assumptions.

Common fluid modeling assumptions:

• incompressible versus compressible
• inviscid versus viscous
• steady versus transient
• laminar versus turbulent
• fully developed versus developing
• internal versus external flow

Dimensionless groups such as Reynolds, Prandtl, Nusselt, Mach, Froude, Biot, and Fourier numbers identify which effects dominate and which correlations are appropriate.

Minor losses represent fittings, entrances, exits, bends, valves, expansions, and contractions:

$$h_m=K\frac{V^2}{2g}$$

External aerodynamic forces are commonly nondimensionalized as:

$$D=\frac{1}{2}\rho V^2{C}_{D}A$$$$L=\frac{1}{2}\rho V^2{C}_{L}A$$

Compressibility becomes important when density changes affect the flow. Mach number is:

$$M=\frac{V}{a}$$

where $a$ is the speed of sound.