Formula Sheet: Transient Systems
Lumped Capacitance Analysis
Biot Number
The Biot number (Bi) determines the applicability of lumped capacitance analysis:
\[\text{Bi} = \frac{hL_c}{k}\]- h = convective heat transfer coefficient (W/m²·K or Btu/hr·ft²·°F)
- Lc = characteristic length (m or ft)
- k = thermal conductivity of the solid (W/m·K or Btu/hr·ft·°F)
Characteristic Length:
\[L_c = \frac{V}{A_s}\]- V = volume of the solid (m³ or ft³)
- As = surface area of the solid (m² or ft²)
Decision Rule: Lumped capacitance analysis is valid when Bi <>. This indicates negligible internal temperature gradients.
Lumped Capacitance Temperature Response
General transient temperature equation:
\[\frac{T(t) - T_\infty}{T_i - T_\infty} = e^{-\frac{t}{\tau}}\]- T(t) = temperature at time t (K or °F)
- T∞ = ambient or fluid temperature (K or °F)
- Ti = initial temperature (K or °F)
- t = time (s or hr)
- τ = time constant (s or hr)
Time Constant:
\[\tau = \frac{\rho V c_p}{hA_s} = \frac{\rho c_p L_c}{h}\]- ρ = density of the solid (kg/m³ or lbm/ft³)
- cp = specific heat capacity (J/kg·K or Btu/lbm·°F)
- V = volume (m³ or ft³)
- As = surface area (m² or ft²)
Alternative form with time constant:
\[T(t) = T_\infty + (T_i - T_\infty)e^{-t/\tau}\]Heat Transfer in Lumped Systems
Instantaneous heat transfer rate:
\[q(t) = hA_s[T(t) - T_\infty] = hA_s(T_i - T_\infty)e^{-t/\tau}\]- q(t) = heat transfer rate at time t (W or Btu/hr)
Total energy transferred from time 0 to t:
\[Q = \rho V c_p (T_i - T(t)) = mc_p(T_i - T(t))\]- Q = total energy transferred (J or Btu)
- m = mass of the solid (kg or lbm)
Maximum possible energy transfer (as t → ∞):
\[Q_{max} = \rho V c_p (T_i - T_\infty) = mc_p(T_i - T_\infty)\]Semi-Infinite Solid Analysis
Applicability
A semi-infinite solid is applicable when:
- The solid extends to infinity in one or more directions
- Temperature changes have not penetrated significantly into the solid
- The boundary opposite the heated/cooled surface remains at initial temperature
Temperature Distribution - Constant Surface Temperature
For sudden change to constant surface temperature Ts:
\[\frac{T(x,t) - T_i}{T_s - T_i} = \text{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right) = 1 - \text{erf}\left(\frac{x}{2\sqrt{\alpha t}}\right)\]- T(x,t) = temperature at position x and time t (K or °F)
- Ts = constant surface temperature (K or °F)
- Ti = initial uniform temperature (K or °F)
- x = distance from the surface (m or ft)
- α = thermal diffusivity (m²/s or ft²/hr)
- erf = error function
- erfc = complementary error function = 1 - erf
Thermal diffusivity:
\[\alpha = \frac{k}{\rho c_p}\]- k = thermal conductivity (W/m·K or Btu/hr·ft·°F)
- ρ = density (kg/m³ or lbm/ft³)
- cp = specific heat (J/kg·K or Btu/lbm·°F)
Heat Flux - Constant Surface Temperature
Surface heat flux:
\[q''_s(t) = \frac{k(T_s - T_i)}{\sqrt{\pi \alpha t}}\]- q''s(t) = surface heat flux at time t (W/m² or Btu/hr·ft²)
Total energy transferred per unit area from time 0 to t:
\[Q'' = 2k(T_s - T_i)\sqrt{\frac{t}{\pi \alpha}}\]- Q'' = energy per unit area (J/m² or Btu/ft²)
Temperature Distribution - Constant Surface Heat Flux
For constant surface heat flux q''s:
\[T(x,t) - T_i = \frac{q''_s}{k}\left[2\sqrt{\frac{\alpha t}{\pi}}\exp\left(-\frac{x^2}{4\alpha t}\right) - x\cdot\text{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right)\right]\]Surface temperature (x = 0):
\[T_s(t) - T_i = \frac{2q''_s\sqrt{\alpha t}}{\sqrt{\pi}k}\]Temperature Distribution - Convection Boundary
For convection at surface with fluid at T∞:
\[\frac{T(x,t) - T_i}{T_\infty - T_i} = \text{erfc}\left(\frac{x}{2\sqrt{\alpha t}}\right) - \exp\left(\frac{hx}{k} + \frac{h^2\alpha t}{k^2}\right)\text{erfc}\left(\frac{x}{2\sqrt{\alpha t}} + \frac{h\sqrt{\alpha t}}{k}\right)\]Surface temperature (x = 0):
\[\frac{T_s(t) - T_i}{T_\infty - T_i} = 1 - \exp\left(\frac{h^2\alpha t}{k^2}\right)\text{erfc}\left(\frac{h\sqrt{\alpha t}}{k}\right)\]Surface heat flux:
\[q''_s(t) = h(T_\infty - T_i)\exp\left(\frac{h^2\alpha t}{k^2}\right)\text{erfc}\left(\frac{h\sqrt{\alpha t}}{k}\right)\]One-Dimensional Transient Conduction - Exact Solutions
Fourier Number
The Fourier number (Fo) is a dimensionless time parameter:
\[\text{Fo} = \frac{\alpha t}{L^2}\]- α = thermal diffusivity (m²/s or ft²/hr)
- t = time (s or hr)
- L = characteristic length (m or ft)
Interpretation: Fo represents the ratio of heat conduction rate to thermal energy storage rate.
General Solution Form
For plane wall, cylinder, or sphere with convection boundary:
\[\theta^* = \frac{T(x,t) - T_\infty}{T_i - T_\infty} = \sum_{n=1}^{\infty} C_n \exp(-\lambda_n^2 \text{Fo}) \cdot f_n(x^*)\]- θ* = dimensionless temperature
- Cn = coefficient dependent on eigenvalue
- λn = eigenvalues (roots of transcendental equation)
- fn(x*) = spatial function (sine, Bessel, etc.)
- x* = dimensionless position
One-Term Approximation (Fo > 0.2)
For Fo > 0.2, the infinite series can be approximated by the first term:
\[\theta^* = C_1 \exp(-\lambda_1^2 \text{Fo}) \cdot f_1(x^*)\]This approximation is typically accurate within a few percent.
Plane Wall (Slab)
Geometry: Thickness 2L, symmetric cooling/heating from both sides
Dimensionless temperature:
\[\theta^* = \frac{T(x,t) - T_\infty}{T_i - T_\infty} = C_1 \exp(-\lambda_1^2 \text{Fo}) \cos(\lambda_1 x^*)\]- x* = x/L (dimensionless position, -1 ≤ x* ≤ 1)
- L = half-thickness of the slab (m or ft)
Centerline temperature (x = 0):
\[\theta_0^* = \frac{T_0 - T_\infty}{T_i - T_\infty} = C_1 \exp(-\lambda_1^2 \text{Fo})\]Temperature at any position:
\[\frac{T(x,t) - T_\infty}{T_0 - T_\infty} = \cos(\lambda_1 x^*)\]Eigenvalue equation:
\[\lambda_1 \tan(\lambda_1) = \text{Bi}\]- Bi = hL/k (Biot number based on half-thickness)
Coefficient C1:
\[C_1 = \frac{4\sin(\lambda_1)}{2\lambda_1 + \sin(2\lambda_1)}\]Infinite Cylinder
Geometry: Radius ro, radial conduction only
Dimensionless temperature:
\[\theta^* = \frac{T(r,t) - T_\infty}{T_i - T_\infty} = C_1 \exp(-\lambda_1^2 \text{Fo}) J_0(\lambda_1 r^*)\]- r* = r/ro (dimensionless radial position, 0 ≤ r* ≤ 1)
- ro = outer radius (m or ft)
- J0 = Bessel function of the first kind, order zero
Centerline temperature (r = 0):
\[\theta_0^* = \frac{T_0 - T_\infty}{T_i - T_\infty} = C_1 \exp(-\lambda_1^2 \text{Fo})\]Temperature at any radius:
\[\frac{T(r,t) - T_\infty}{T_0 - T_\infty} = J_0(\lambda_1 r^*)\]Eigenvalue equation:
\[\lambda_1 \frac{J_1(\lambda_1)}{J_0(\lambda_1)} = \text{Bi}\]- Bi = hro/k (Biot number based on radius)
- J1 = Bessel function of the first kind, order one
Coefficient C1:
\[C_1 = \frac{2J_1(\lambda_1)}{\lambda_1[J_0^2(\lambda_1) + J_1^2(\lambda_1)]}\]Sphere
Geometry: Radius ro
Dimensionless temperature:
\[\theta^* = \frac{T(r,t) - T_\infty}{T_i - T_\infty} = C_1 \exp(-\lambda_1^2 \text{Fo}) \frac{\sin(\lambda_1 r^*)}{\lambda_1 r^*}\]- r* = r/ro (dimensionless radial position, 0 ≤ r* ≤ 1)
- ro = outer radius (m or ft)
Center temperature (r = 0):
\[\theta_0^* = \frac{T_0 - T_\infty}{T_i - T_\infty} = C_1 \exp(-\lambda_1^2 \text{Fo})\]Temperature at any radius:
\[\frac{T(r,t) - T_\infty}{T_0 - T_\infty} = \frac{\sin(\lambda_1 r^*)}{\lambda_1 r^*}\]Eigenvalue equation:
\[1 - \lambda_1 \cot(\lambda_1) = \text{Bi}\]- Bi = hro/k (Biot number based on radius)
Coefficient C1:
\[C_1 = \frac{4[\sin(\lambda_1) - \lambda_1\cos(\lambda_1)]}{2\lambda_1 - \sin(2\lambda_1)}\]Total Energy Transfer
Dimensionless energy transfer for all three geometries:
\[\frac{Q}{Q_0} = 1 - \frac{Q_{max} - Q}{Q_{max}} = 1 - \theta_0^* \cdot S\]- Q = energy transferred up to time t (J or Btu)
- Q0 or Qmax = maximum possible energy transfer = ρVcp(Ti - T∞)
- S = shape factor for energy
Shape factors (one-term approximation):
- Plane wall: \(S = \frac{\sin(\lambda_1)}{\lambda_1}\)
- Infinite cylinder: \(S = \frac{2J_1(\lambda_1)}{\lambda_1}\)
- Sphere: \(S = 3\frac{\sin(\lambda_1) - \lambda_1\cos(\lambda_1)}{\lambda_1^3}\)
Multidimensional Systems - Product Solution
Product Solution Method
For multidimensional geometries formed by intersection of one-dimensional solutions:
The dimensionless temperature is the product of the corresponding one-dimensional solutions.
Two-Dimensional Systems
Infinite rectangular bar (plane wall × plane wall):
\[\theta^*(x,y,t) = \theta^*_{wall,x}(x,t) \cdot \theta^*_{wall,y}(y,t)\]Semi-infinite cylinder (infinite cylinder × plane wall):
\[\theta^*(r,x,t) = \theta^*_{cyl}(r,t) \cdot \theta^*_{wall}(x,t)\]Three-Dimensional Systems
Rectangular parallelepiped (plane wall × plane wall × plane wall):
\[\theta^*(x,y,z,t) = \theta^*_{wall,x}(x,t) \cdot \theta^*_{wall,y}(y,t) \cdot \theta^*_{wall,z}(z,t)\]Short cylinder (infinite cylinder × plane wall):
\[\theta^*(r,x,t) = \theta^*_{cyl}(r,t) \cdot \theta^*_{wall}(x,t)\]- Infinite cylinder solution in radial direction
- Plane wall solution in axial direction (thickness 2L)
Energy Transfer in Multidimensional Systems
For two-dimensional systems:
\[\frac{Q}{Q_{max}} = \left(\frac{Q}{Q_{max}}\right)_1 + \left(\frac{Q}{Q_{max}}\right)_2 \left[1 - \left(\frac{Q}{Q_{max}}\right)_1\right]\]For three-dimensional systems:
\[\frac{Q}{Q_{max}} = \left(\frac{Q}{Q_{max}}\right)_1 + \left(\frac{Q}{Q_{max}}\right)_2 \left[1 - \left(\frac{Q}{Q_{max}}\right)_1\right] + \left(\frac{Q}{Q_{max}}\right)_3 \left[1 - \left(\frac{Q}{Q_{max}}\right)_1\right]\left[1 - \left(\frac{Q}{Q_{max}}\right)_2\right]\]Where subscripts 1, 2, 3 refer to the individual one-dimensional geometries comprising the multidimensional system.
Transient Conduction with Heat Generation
Plane Wall with Uniform Heat Generation
Energy balance equation:
\[\rho c_p \frac{\partial T}{\partial t} = k\frac{\partial^2 T}{\partial x^2} + \dot{q}'''\]- q̇''' = volumetric heat generation rate (W/m³ or Btu/hr·ft³)
Steady-state temperature distribution (symmetric, thickness 2L):
\[T(x) = T_s + \frac{\dot{q}'''L^2}{2k}\left[1 - \left(\frac{x}{L}\right)^2\right]\]- Ts = surface temperature (K or °F)
- x = distance from centerline (m or ft)
- L = half-thickness (m or ft)
Maximum temperature (at centerline, x = 0):
\[T_{max} = T_s + \frac{\dot{q}'''L^2}{2k}\]Cylinder with Uniform Heat Generation
Steady-state radial temperature distribution:
\[T(r) = T_s + \frac{\dot{q}'''r_o^2}{4k}\left[1 - \left(\frac{r}{r_o}\right)^2\right]\]- r = radial position (m or ft)
- ro = outer radius (m or ft)
Maximum temperature (at centerline, r = 0):
\[T_{max} = T_s + \frac{\dot{q}'''r_o^2}{4k}\]Sphere with Uniform Heat Generation
Steady-state radial temperature distribution:
\[T(r) = T_s + \frac{\dot{q}'''r_o^2}{6k}\left[1 - \left(\frac{r}{r_o}\right)^2\right]\]Maximum temperature (at center, r = 0):
\[T_{max} = T_s + \frac{\dot{q}'''r_o^2}{6k}\]Numerical Methods for Transient Conduction
Finite Difference Formulation - Explicit Method
One-dimensional transient heat conduction (interior node):
\[\frac{T_m^{p+1} - T_m^p}{\Delta t} = \alpha \frac{T_{m+1}^p - 2T_m^p + T_{m-1}^p}{(\Delta x)^2}\]- Tmp = temperature at node m and time step p
- Δt = time step (s or hr)
- Δx = spatial step (m or ft)
Rearranged for explicit solution:
\[T_m^{p+1} = \text{Fo}(T_{m+1}^p + T_{m-1}^p) + (1 - 2\text{Fo})T_m^p\]Where Fo = αΔt/(Δx)²
Stability Criterion - Explicit Method
For one-dimensional explicit finite difference:
\[\text{Fo} = \frac{\alpha \Delta t}{(\Delta x)^2} \leq \frac{1}{2}\]This ensures numerical stability. Violation leads to divergent, non-physical solutions.
Maximum allowable time step:
\[\Delta t \leq \frac{(\Delta x)^2}{2\alpha}\]Finite Difference - Surface Node with Convection
Energy balance at surface node (m = 0) with convection:
\[\rho c_p \frac{\Delta x}{2} \frac{T_0^{p+1} - T_0^p}{\Delta t} = k\frac{T_1^p - T_0^p}{\Delta x} + h(T_\infty - T_0^p)\]Rearranged for explicit solution:
\[T_0^{p+1} = 2\text{Fo}(T_1^p + \text{Bi} \cdot T_\infty) + (1 - 2\text{Fo} - 2\text{Fo} \cdot \text{Bi})T_0^p\]- Bi = hΔx/k
Stability criterion for surface node:
\[\text{Fo}(1 + \text{Bi}) \leq \frac{1}{2}\]Implicit Method (Fully Implicit)
Interior node formulation:
\[-\text{Fo} \cdot T_{m-1}^{p+1} + (1 + 2\text{Fo})T_m^{p+1} - \text{Fo} \cdot T_{m+1}^{p+1} = T_m^p\]This method is unconditionally stable for all values of Fo, but requires solving a system of simultaneous equations at each time step.
Crank-Nicolson Method
Average of explicit and implicit methods:
\[T_m^{p+1} = T_m^p + \frac{\text{Fo}}{2}\left[(T_{m+1}^{p+1} - 2T_m^{p+1} + T_{m-1}^{p+1}) + (T_{m+1}^p - 2T_m^p + T_{m-1}^p)\right]\]This method is also unconditionally stable and generally more accurate than purely explicit or implicit methods.
Transient Diffusion in Binary Systems
Fick's Second Law
Unsteady-state diffusion equation:
\[\frac{\partial C_A}{\partial t} = D_{AB}\frac{\partial^2 C_A}{\partial x^2}\]- CA = concentration of species A (kmol/m³ or lb-mol/ft³)
- t = time (s or hr)
- DAB = mass diffusivity of A in B (m²/s or ft²/hr)
- x = position (m or ft)
Note: This equation is mathematically analogous to the heat conduction equation, allowing use of identical solutions with appropriate substitutions.
Analogy Between Heat and Mass Transfer
Parameter correspondence:
- Temperature T ↔ Concentration CA
- Thermal diffusivity α ↔ Mass diffusivity DAB
- Thermal conductivity k ↔ DAB
- Convective heat transfer coefficient h ↔ Mass transfer coefficient kc
Semi-Infinite Medium - Mass Transfer
Constant surface concentration:
\[\frac{C_A(x,t) - C_{A,i}}{C_{A,s} - C_{A,i}} = \text{erfc}\left(\frac{x}{2\sqrt{D_{AB}t}}\right)\]- CA,i = initial uniform concentration
- CA,s = constant surface concentration
Surface mass flux:
\[n''_{A,s}(t) = \frac{D_{AB}(C_{A,s} - C_{A,i})}{\sqrt{\pi D_{AB}t}}\]- n''A,s = mass flux of species A at surface (kmol/m²·s or lb-mol/ft²·hr)
Penetration Depth
Thermal penetration depth (approximate):
\[\delta_t \approx 2\sqrt{\alpha t}\]Mass transfer penetration depth (approximate):
\[\delta_m \approx 2\sqrt{D_{AB}t}\]These represent the approximate distance to which temperature or concentration changes have penetrated into the medium.
Special Cases and Applications
Periodic Heating/Cooling
Surface temperature variation:
\[T_s(t) = T_m + T_a\sin(\omega t)\]- Tm = mean surface temperature (K or °F)
- Ta = amplitude of temperature variation (K or °F)
- ω = angular frequency = 2π/τp (rad/s)
- τp = period of oscillation (s)
Temperature response in semi-infinite medium:
\[T(x,t) = T_m + T_a e^{-x\sqrt{\omega/(2\alpha)}}\sin\left(\omega t - x\sqrt{\frac{\omega}{2\alpha}}\right)\]Thermal wave characteristics:
- Attenuation: Amplitude decreases exponentially with depth
- Phase lag: Temperature variations lag behind surface by time \(\Delta t = x\sqrt{\frac{1}{2\alpha\omega}}\)
Contact Between Two Semi-Infinite Solids
Interface temperature when two semi-infinite solids at different initial temperatures TA,i and TB,i are brought into contact:
\[T_{interface} = \frac{\sqrt{k_A\rho_A c_{p,A}} \cdot T_{A,i} + \sqrt{k_B\rho_B c_{p,B}} \cdot T_{B,i}}{\sqrt{k_A\rho_A c_{p,A}} + \sqrt{k_B\rho_B c_{p,B}}}\]Or equivalently:
\[T_{interface} = \frac{(k\rho c_p)_A^{1/2} T_{A,i} + (k\rho c_p)_B^{1/2} T_{B,i}}{(k\rho c_p)_A^{1/2} + (k\rho c_p)_B^{1/2}}\]- (kρcp)1/2 = thermal effusivity or thermal inertia
Time to Reach Specific Temperature
For lumped capacitance systems, time to reach temperature T:
\[t = -\tau \ln\left(\frac{T - T_\infty}{T_i - T_\infty}\right)\]For one-dimensional systems using charts/tables (Fo > 0.2):
- Calculate Biot number
- Determine desired dimensionless temperature θ*
- Use Heisler/Grober charts or tables to find Fourier number
- Calculate time: \(t = \frac{\text{Fo} \cdot L^2}{\alpha}\)
