PE Exam Exam > PE Exam Notes > Chemical Engineering for PE > Cheatsheet: Transient Systems
Cheatsheet: Transient Systems
1. Lumped Capacitance Method
1.1 Fundamental Criterion
| Parameter | Description |
|---|---|
| Biot Number (Bi) | Bi = hLc/k where Lc = V/As (characteristic length) |
| Validity Condition | Bi < 0.1="" for="" lumped="" capacitance="" to="" be="" valid="" (internal="" resistance="">< external=""> |
| Physical Meaning | Temperature variation within solid is negligible compared to difference between surface and surroundings |
1.2 Governing Equation
| Equation | Variables |
|---|---|
| ρVCp(dT/dt) = -hAs(T - T∞) | ρ = density, V = volume, Cp = specific heat, h = convection coefficient, As = surface area |
| (T - T∞)/(Ti - T∞) = exp(-t/τ) | τ = ρVCp/(hAs) = thermal time constant |
| Q = ρVCp(Ti - T) | Q = total heat transfer up to time t |
1.3 Characteristic Length for Common Geometries
| Geometry | Lc = V/As |
|---|---|
| Plane Wall (thickness 2L) | Lc = L |
| Long Cylinder (radius ro) | Lc = ro/2 |
| Sphere (radius ro) | Lc = ro/3 |
| Cube (side L) | Lc = L/6 |
2. One-Dimensional Transient Conduction
2.1 Governing Parameters
| Parameter | Definition |
|---|---|
| Fourier Number (Fo) | Fo = αt/Lc2 where α = k/(ρCp) = thermal diffusivity; dimensionless time |
| Biot Number (Bi) | Bi = hLc/k; ratio of internal to external thermal resistance |
| Temperature Ratio (θ) | θ = (T - T∞)/(Ti - T∞) |
2.2 Plane Wall Solution
| Expression | Description |
|---|---|
| θ(x,t) = Σ Cnexp(-λn2Fo)cos(λnx/L) | Infinite series solution; λn are eigenvalues from λntan(λn) = Bi |
| θo(t) = C1exp(-λ12Fo) | One-term approximation for centerline (Fo > 0.2); C1 from tables/charts |
| θ(x,t) = θo(t)cos(λ1x/L) | Spatial variation for one-term approximation |
2.3 Infinite Cylinder Solution
| Expression | Description |
|---|---|
| θ(r,t) = Σ Cnexp(-λn2Fo)J0(λnr/ro) | Infinite series; λn from λnJ1(λn)/J0(λn) = Bi |
| θo(t) = C1exp(-λ12Fo) | One-term approximation for centerline (Fo > 0.2) |
| θ(r,t) = θo(t)J0(λ1r/ro)/J0(λ1) | Spatial variation; J0 is Bessel function of first kind, order zero |
2.4 Sphere Solution
| Expression | Description |
|---|---|
| θ(r,t) = Σ Cnexp(-λn2Fo)sin(λnr/ro)/(λnr/ro) | Infinite series; λn from 1 - λncot(λn) = Bi |
| θo(t) = C1exp(-λ12Fo) | One-term approximation for center (Fo > 0.2) |
| θ(r,t) = θo(t)sin(λ1r/ro)/(λ1r/ro) | Spatial variation for one-term approximation |
2.5 Total Energy Transfer
| Geometry | Q/Qmax |
|---|---|
| Plane Wall | Q/Qmax = 1 - θosin(λ1)/λ1 |
| Cylinder | Q/Qmax = 1 - 2θoJ1(λ1)/λ1 |
| Sphere | Q/Qmax = 1 - 3θo[sin(λ1) - λ1cos(λ1)]/λ13 |
| Qmax | Qmax = ρVCp(Ti - T∞) |
3. Semi-Infinite Solid
3.1 Defining Characteristics
- Extends to infinity in one or more directions
- Valid when penetration depth is small compared to solid thickness
- Penetration depth δ ≈ 2√(αt)
- Appropriate when Fo = αt/L2 <>
3.2 Boundary Condition Solutions
| Boundary Condition | Temperature Distribution |
|---|---|
| Constant Surface Temperature Ts | (T - Ti)/(Ts - Ti) = erfc(x/(2√(αt))) = 1 - erf(x/(2√(αt))) |
| Constant Surface Heat Flux q"s | (T - Ti) = (2q"s/k)√(αt/π)exp(-x2/(4αt)) - (q"sx/k)erfc(x/(2√(αt))) |
| Convection at Surface | (T - Ti)/(T∞ - Ti) = erfc(x/(2√(αt))) - exp(hx/k + h2αt/k2)erfc(x/(2√(αt)) + h√(αt)/k) |
3.3 Surface Heat Flux and Energy Transfer
| Boundary Condition | Surface Heat Flux q"s(t) |
|---|---|
| Constant Surface Temperature | q"s(t) = k(Ts - Ti)/√(παt) |
| Convection at Surface | q"s(t) = h(T∞ - Ti)exp(h2αt/k2)erfc(h√(αt)/k) |
3.4 Error Function Properties
| Function | Definition/Property |
|---|---|
| erf(η) | erf(η) = (2/√π)∫0ηexp(-u2)du |
| erfc(η) | erfc(η) = 1 - erf(η) |
| erf(0) | 0 |
| erf(∞) | 1 |
| erf(-η) | -erf(η) |
4. Multidimensional Systems
4.1 Product Solution Method
| Configuration | Solution |
|---|---|
| Two-Dimensional (e.g., short cylinder) | θ(x,y,t) = θ1D,x(x,t) × θ1D,y(y,t) |
| Three-Dimensional (e.g., rectangular parallelepiped) | θ(x,y,z,t) = θ1D,x(x,t) × θ1D,y(y,t) × θ1D,z(z,t) |
| Energy Transfer | (Q/Qmax)2D = (Q/Qmax)x + (Q/Qmax)y[1 - (Q/Qmax)x] |
4.2 Common Multidimensional Geometries
| Geometry | Product Solution Components |
|---|---|
| Short Cylinder (radius ro, length 2L) | Infinite cylinder × infinite plane wall |
| Semi-Infinite Cylinder | Infinite cylinder × semi-infinite solid |
| Rectangular Bar (2L1 × 2L2 × ∞) | Plane wall (2L1) × plane wall (2L2) |
| Rectangular Parallelepiped (2L1 × 2L2 × 2L3) | Plane wall (2L1) × plane wall (2L2) × plane wall (2L3) |
| Semi-Infinite Plate | Plane wall × semi-infinite solid |
| Quarter-Infinite Solid | Semi-infinite solid × semi-infinite solid |
4.3 Solution Procedure
- Identify constituent one-dimensional geometries
- Determine Bi and Fo for each dimension using appropriate characteristic length
- Find θ for each one-dimensional component from charts/tables/equations
- Multiply individual θ values to obtain multidimensional solution
- Apply product formula for energy transfer calculations
5. Periodic Heating
5.1 Surface Temperature Variation
| Parameter | Expression |
|---|---|
| Surface Temperature | Ts(t) = Tm + Tasin(ωt) |
| ω (angular frequency) | ω = 2π/tp where tp = period |
| Tm | Mean surface temperature |
| Ta | Amplitude of surface temperature oscillation |
5.2 Temperature Distribution in Semi-Infinite Solid
| Expression | Description |
|---|---|
| T(x,t) = Tm + Taexp(-x√(ω/(2α)))sin(ωt - x√(ω/(2α))) | Steady-periodic solution after initial transient dies out |
| δp = √(2α/ω) | Penetration depth; amplitude decreases to exp(-1) ≈ 0.368 at x = δp |
| Phase lag | Δφ = x√(ω/(2α)) radians |
5.3 Applications
- Daily/seasonal temperature variations in soil
- Periodic furnace heating cycles
- Heat treatment processes with cyclic heating
- Building thermal mass analysis with daily temperature swings
6. Finite Difference Methods
6.1 Spatial Discretization
| Method | Expression |
|---|---|
| Central Difference (2nd derivative) | ∂2T/∂x2 ≈ (Tm-1 - 2Tm + Tm+1)/(Δx)2 |
| Forward Difference (1st derivative) | ∂T/∂t ≈ (Tmp+1 - Tmp)/Δt |
| Node Spacing | Δx = L/M where M = number of spacing intervals |
6.2 Explicit Method
| Equation | Description |
|---|---|
| Tmp+1 = Fo(Tm-1p + Tm+1p) + (1 - 2Fo)Tmp | Interior node equation; Fo = αΔt/(Δx)2 |
| Stability Criterion | Fo ≤ 1/2 for one-dimensional problems |
| Advantages | Simple, straightforward calculation; temperature at new time step calculated directly |
| Disadvantages | Stability restriction limits time step size |
6.3 Implicit Method
| Equation | Description |
|---|---|
| -FoTm-1p+1 + (1 + 2Fo)Tmp+1 - FoTm+1p+1 = Tmp | Interior node equation using backward difference in time |
| Stability | Unconditionally stable for all Fo values |
| Solution Method | Requires simultaneous solution of system of algebraic equations (matrix inversion) |
| Advantages | No stability restriction; larger time steps permitted |
6.4 Boundary Conditions
| Boundary Type | Explicit Formulation |
|---|---|
| Convection Surface (node 0) | T0p+1 = 2Fo(T1p + BiT∞) + (1 - 2Fo - 2FoBi)T0p; Stability: Fo(1 + Bi) ≤ 1/2 |
| Insulated Surface | T0p+1 = 2FoT1p + (1 - 2Fo)T0p; Stability: Fo ≤ 1/2 |
| Constant Surface Temperature | T0p+1 = Ts (specified value) |
| Constant Surface Heat Flux | T0p+1 = 2Fo(T1p + q"sΔx/k) + (1 - 2Fo)T0p |
6.5 Cylindrical and Spherical Coordinates
| Geometry | Interior Node (Explicit) |
|---|---|
| Cylinder (radial) | Tmp+1 = Fo[(1 + Δr/(2rm))Tm+1p + (1 - Δr/(2rm))Tm-1p] + (1 - 2Fo)Tmp |
| Sphere (radial) | Tmp+1 = Fo[(1 + Δr/rm)Tm+1p + (1 - Δr/rm)Tm-1p] + (1 - 2Fo)Tmp |
| Fo Definition | Fo = αΔt/(Δr)2 |
7. Solution Charts and Tables
7.1 Heisler Charts
- Graphical solutions for centerline/midplane temperature vs. Fo with Bi as parameter
- Valid for one-dimensional plane wall, cylinder, and sphere geometries
- Chart 1: Centerline temperature θo vs. Fo for various Bi
- Chart 2: Temperature distribution θ/θo vs. position for various Bi and Fo
- Chart 3: Heat transfer Q/Qmax vs. Fo for various Bi
- Accurate for Fo > 0.2 (one-term approximation valid)
7.2 Grober Charts
- Alternative graphical solutions covering full Fo range (0 to ∞)
- Plot θ vs. √Fo with Bi as parameter
- Useful for short-time transients (Fo <>
7.3 Coefficient Tables
| Parameter | Usage |
|---|---|
| λ1 | First eigenvalue; function of Bi from transcendental equations |
| C1 | First coefficient in series solution; function of Bi |
| Tables | Provide λ1 and C1 values for range of Bi for plane wall, cylinder, sphere |
| Application | Used in one-term approximation for Fo > 0.2 |
8. Practical Considerations
8.1 Method Selection Criteria
| Condition | Recommended Method |
|---|---|
| Bi <> | Lumped capacitance method |
| 0.1 < bi,="" fo=""> 0.2, regular geometry | One-term approximation or Heisler charts |
| Fo <> | Infinite series solution or Grober charts |
| Penetration depth <> | Semi-infinite solid solution |
| Irregular geometry or complex BC | Finite difference methods |
| Multidimensional, regular geometry | Product solution method |
8.2 Key Dimensionless Groups
| Group | Physical Significance |
|---|---|
| Bi = hLc/k | Internal resistance/external resistance; determines temperature uniformity within solid |
| Fo = αt/Lc2 | Dimensionless time; heat conducted/heat stored |
| θ = (T - T∞)/(Ti - T∞) | Normalized temperature; ranges from 1 (initial) to 0 (equilibrium) |
8.3 Common Errors to Avoid
- Using lumped capacitance when Bi > 0.1
- Incorrect characteristic length Lc for geometry (V/As, not total dimension)
- Applying one-term approximation for Fo <>
- Violating stability criterion in explicit finite difference (Fo > 1/2)
- Confusing half-thickness L with full thickness 2L in plane wall problems
- Incorrect Bi definition using wrong length scale
8.4 Property Evaluation
- Thermal properties (k, α, ρ, Cp) evaluated at average temperature Tavg = (Ti + T∞)/2
- For large temperature changes, iterate solution with updated properties
- Convection coefficient h may depend on surface temperature; update as needed
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