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![Finite-Difference Methods
Analytical methods solve some 2D steady conduction
problems.
Applicable to simple geometries and boundary conditions.
Solutions are well documented in literature [1–5].
Most 2D problems are too complex for analytical solutions.
Numerical methods (finite-difference, finite-element, boundary-
element) are preferred.
Numerical methods extend easily to 3D problems.
Finite-difference method is simple and ideal for beginners.
Bowman, R. A., A. C. Mueller, and W. M. Nagle, Trans. ASME, 62, 283, 1940.
2Standards of the Tubular Exchange Manufacturers Association, 6th ed., Tubular Exchange Manufacturers Association, New York, 1978.
Jakob, M., Heat Transfer, Vol. 2, Wiley, New York, 1957.
Kays, W. M., and A. L. London, Compact Heat Exchangers, 3rd ed., McGraw-Hill, New York, 1984.
Kakac, S., A. E. Bergles, and F. Mayinger, Eds., Heat Exchangers, Hemisphere Publishing, New York, 1981.](https://image.slidesharecdn.com/transientflow-251108212457-687ce801/75/Transient-flow-in-heat-transfer-Unsteady-state-pptx-43-2048.jpg)
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Transient flow in heat transfer (Unsteady state).pptx
- 1.
- 2. Initial Focus: Began with 1D, steady-state conduction without internal generation. Progression: Included multidimensional effects and internal heat generation. New Focus: Now considering time-dependent (transient) conduction. Unsteady Conditions: Caused by changes in boundary conditions. Example: Hot metal billet exposed to cool air – temp. drops until steady state. Chapter Goal: Determine temperature distribution over time. Analyze heat transfer to/from solid during transient process.
- 3. Method SelectionDepends on Assumptions: Negligible internal temp. gradients → Lumped capacitance method (Sections 5.1–5.3). Significant gradients, 1D conduction → Exact heat equation solutions: Plane walls, cylinders, spheres (Sections 5.4–5.6). Semi-infinite solids (Section 5.7).Special Cases: Step surface changes → Section 5.8. Periodic surface heating → Section 5.9.Complex Cases: Use finite-difference/finite-element methods (Section 5.10).
- 4. The Lumped CapacitanceMethod Problem: Transient conduction in a solid with sudden thermal change. Example: Hot metal forging initially at temperature 𝑇𝑖. Action: Quenched in liquid at lower temperature 𝑇∞ < 𝑇𝑖. Start time: Quenching begins at = 0. 𝑡 Effect: Solid temperature drops over time > 0. 𝑡 Cause: Convection heat transfer at solid–liquid interface. Lumped Capacitance Assumption: Temperature is uniform throughout the solid at any time. Internal temperature gradients are negligible.
- 5. Cooling of ahot metal forging
- 6. The Lumped CapacitanceMethod Fourier’s Law: Heat conduction needs a temperature gradient. No gradient → Implies infinite thermal conductivity (not realistic). Approximation: Valid if internal conduction resistance external ≪ convection resistance. Assumption: Internal resistance is negligible for current analysis.
- 7. The Lumped CapacitanceMethod No internal gradients → Heat equation not applicable.Heat equation: Governs spatial temperature distribution. Alternative: Use overall energy balance on the solid. Goal: Relate surface heat loss to internal energy change. Approach: Apply Equation to control volume (Figure (slide 5)).
- 8. The Lumped CapacitanceMethod Equation : Used to determine the time required for the solid to reach a specific temperature T. Equation : Used to compute the temperature reached by the solid at a given time t. Temperature difference between solid and fluid decays exponentially as t → ∞. This behavior is shown in Figure (next slide). From Equation , the term (ρVc/hAs) represents the thermal time constant.
- 9. Transient temperature responseof lumped capacitance solids for different thermal time constants .
- 10. The Lumped CapacitanceMethod From Equation : Rt = convection thermal resistance Ct = lumped thermal capacitance Higher Rt or Ct → slower thermal response Analogous to voltage decay in an electrical RC circuit To determine the total energy transfer Q occurring up to some time t, we simply write
- 11. The Lumped CapacitanceMethod Substituting for θ from Equation and integrating, we obtain The quantity Q is, of course, related to the change in the internal energy of the solid, and from Equation For quenching: Q > 0 → solid loses energy For heating (θ < 0): Q < 0 → solid gains energy Equations 5.5, 5.6, 5.8a apply to both cases.
- 12. Validity of theLumped Capacitance Method Lumped capacitance method: Simple and convenient for transient heat/cool problems This section: Defines conditions for accurate use of the method
- 13. Validity of theLumped Capacitance Method Consider steady-state conduction through a wall of area A (Figure next slide) Criterion applies to transient cases as well One surface at Ts,1, other exposed to fluid at T∞ < Ts,1 Opposite surface reaches Ts,2, where T∞ < Ts,2 < Ts,1 Under steady-state, surface energy balance (Eq. 1.13) simplifies to: Reduced form of Equation applies
- 14. Validity of theLumped Capacitance Method where k is the thermal conductivity of the solid. Rearranging, we then obtain Effect of Biot number on steady-state temperature distribution in a plane wall with surface convection.
- 15. Validity of theLumped Capacitance Method (hL/k) in Equation = Biot number (Bi) is dimensionless; key in conduction with convection Bi measures: Temp drop within solid vs. solid–fluid temp difference Bi = ratio of thermal resistances (solid vs. fluid) If Bi 1: ≪ Conduction resistance convection resistance→ Uniform ≪ temperature in solid is a valid assumption
- 16. Validity of theLumped Capacitance Method Lumped capacitance method: simple, convenient for transient heat transfer. Useful for heating and cooling analysis. Next: define conditions for accurate application.
- 17. Effect of Biotnumber on steady- state temperature distribution in a plane wall with surface convection.
- 18. Validity of theLumped Capacitance Method Develop criterion via steady-state conduction through plane wall (area A). Criterion extends to transient processes. One surface at temperature Ts,1. Other surface exposed to fluid at T∞ (T∞ < Ts,1). That surface reaches intermediate temperature Ts,2 (T∞ < Ts,2 < Ts,1). Apply surface energy balance (Equation ) under steady-state.
- 19. Validity of theLumped Capacitance Method where k is the thermal conductivity of the solid. Rearranging, we then obtain
- 20. Validity of theLumped Capacitance Method Biot number (Bi = hL/k) is dimensionless. Key in conduction with surface convection. Measures temperature drop inside solid vs. surface-fluid difference. Also represents ratio of thermal resistances: Conduction (solid) vs. convection (fluid). If Bi << 1: Conduction resistance convection resistance. ≪ Uniform temperature in solid is a valid assumption.
- 21. Validity of theLumped Capacitance Method Biot number also key in transient conduction. Consider a plane wall: Initial temp Ti, fluid temp T∞ < Ti. Cooling by convection, 1D heat transfer. Temperature T(x, t) depends strongly on Bi. Bi << 1: Small internal gradients. T(x, t) ≈ T(t) → uniform solid temperature. Temp drop mainly at surface-fluid interface.
- 22. Validity of theLumped Capacitance Method Bi ≥ 1: Significant internal gradients. Must use T = T(x, t). Bi >> 1: Large temp drop inside solid. Small drop across fluid layer.
- 23. Validity of theLumped Capacitance Method Biot number relevance: Important in transient conduction problems. Setup: Plane wall, initially at Ti, exposed to fluid at T∞ < Ti. Assumption: One-dimensional heat flow in x-direction. Goal: Analyze T(x, t) — temperature variation over position and time. Bi << 1: Small internal temperature gradients. Approximate as T(x, t) ≈ T(t) (uniform solid temperature). Major temperature drop across fluid–solid interface.
- 24. Validity of theLumped Capacitance Method Moderate to high Bi: Significant internal temperature gradients. Must solve T = T(x,t). Bi >> 1: Large temperature difference within solid. Smaller difference between surface and fluid.
- 25. Validity of theLumped Capacitance Method Lumped capacitance method: Preferred for transient heating/cooling. Known for simplicity. First step: Calculate Biot number (Bi). If Bi ≤ 0.1: Lumped method is valid. Use T(t) instead of T(x, t).
- 26. Validity of theLumped Capacitance Method Transient temperature distributions for different Biot numbers in a plane wall symmetrically cooled by convection.
- 27. Validity of theLumped Capacitance Method Lumped method error: Small if Bi ≤ 0.1. Characteristic length (Lc): Defined as Lc = V / As (volume/surface area). Simplifies shape handling. Special cases: Plane wall (thickness 2L): Lc = L. Long cylinder: Lc = ro / 2. Sphere: Lc = ro / 3.
- 28. Validity of theLumped Capacitance Method Finally, we note that, with Lc ≡ V/As, the exponent of Equation may be expressed as It is a dimensionless time, which, with the Biot number, characterizes transient conduction problems.
- 34. General Lumped Capacitance Analysis Transient conduction often starts from convection with an adjoining fluid. Other causes include: Radiation: from large surroundings across gas or vacuum. Surface heat flux: applied to part or all of the surface. Internal heat generation: within the solid itself. Examples: Surface heating via sheet/film electrical heater. Internal heating via electrical current through the solid.
- 35. General Lumped Capacitance Analysis Figure: Shows combined effects on a solid's thermal state from: Convection Radiation Surface heat flux 𝑞𝑠 ” Internal energy generation ˙ 𝑞 Control surface for general lumped capacitance analysis
- 36. General Lumped Capacitance Analysis Initial condition (at =0): 𝑡 Solid temperature 𝑇𝑖 ≠ fluid temperature 𝑇∞ and surroundings 𝑠𝑢𝑟 Heating sources initiated: Surface heat flux 𝑞𝑠′′ Volumetric generation ˙ 𝑞 Surface interactions: Heat flux 𝑞𝑠′′ on surface area 𝐴 ( ) 𝑠 ℎ Convection & radiation on 𝐴 ( , ) 𝑠 𝑐 𝑟 Convection-radiation assumed from surface only Surfaces may differ: , ≠ , 𝐴𝑠 𝑐 𝐴𝑠 𝑟 Energy conservation applied at any time t using Equation + 𝑡
- 37. General Lumped Capacitance Analysis Energyconservation applied at any time t using Equation + 𝑡 or, from Equations and ,
- 38. General Lumped Capacitance Analysis:Radiation Only If there is no imposed heat flux or generation and convection is either nonexistent (a vacuum) or negligible relative to radiation, Equation reduces to Separating variables and integrating from the initial condition to any time t, it follows that
- 39. General Lumped Capacitance Analysis:Radiation Only Evaluating both integrals and rearranging, the time required to reach the temperature T becomes
- 40. General Lumped Capacitance Analysis:Radiation Only Expression estimates time for solid to reach temperature Equation 5.17 can be integrated. Limiting caseT. : Tsur = 0 (radiation to deep space). Yields simplified result for this condition.
- 41. General Lumped Capacitance Analysis:Negligible Radiation Exact solution possible if:Radiation neglected All variables (except T) time-independent Define: Temperature difference: θ ≡ T − T∞ Time derivative: dθ/dt = dT/dt Result: Equation reduces becomes linear, first-order, nonhomogeneous ODE
- 42. General Lumped Capacitance Analysis:Negligible Radiation
- 43. Finite-Difference Methods Analyticalmethods solve some 2D steady conduction problems. Applicable to simple geometries and boundary conditions. Solutions are well documented in literature [1–5]. Most 2D problems are too complex for analytical solutions. Numerical methods (finite-difference, finite-element, boundary- element) are preferred. Numerical methods extend easily to 3D problems. Finite-difference method is simple and ideal for beginners. Bowman, R. A., A. C. Mueller, and W. M. Nagle, Trans. ASME, 62, 283, 1940. 2Standards of the Tubular Exchange Manufacturers Association, 6th ed., Tubular Exchange Manufacturers Association, New York, 1978. Jakob, M., Heat Transfer, Vol. 2, Wiley, New York, 1957. Kays, W. M., and A. L. London, Compact Heat Exchangers, 3rd ed., McGraw-Hill, New York, 1984. Kakac, S., A. E. Bergles, and F. Mayinger, Eds., Heat Exchangers, Hemisphere Publishing, New York, 1981.
- 44. Finite-Difference Methods Analyticalsolutions suit simple geometries and boundary conditions (e.g., 1D cases). Some 2D/3D cases allow analytical solutions. Complex geometries or boundaries require numerical methods. Finite-difference/element methods are alternatives. This section covers explicit and implicit finite-difference methods for transient conduction.
- 45. Discretization of theHeat Equation: The Explicit Method Consider the 2D system from Figure (Below). Conditions: transient, constant properties, no internal heat generation. Use heat equation form from Equation ++.
- 46. Discretization of theHeat Equation: The Explicit Method Use central-difference approximations for spatial derivatives (Eqs. & ). Subscripts , : x- and y-direction nodal positions. 𝑚 𝑛 Discretize both space and time. Introduce integer : time step index. 𝑝
- 47. Discretization of theHeat Equation: The Explicit Method The finite-difference approximation to the time derivative in Equation is expressed as Superscript : denotes time level of temperature 𝑝 𝑇 Time derivative: approximated by difference between 𝑇 +1 𝑝 and Calculations 𝑇𝑝 done step-by-step at time intervals ΔT Like space, temperature is computed only at discrete time points Solution restricted to grid points in space and time
- 49. Discretization of theHeat Equation: The Explicit Method Eq. =→ substituted into Eq. Solution form depends on time level used for spatial derivatives Explicit method: evaluates temperatures at previous time 𝑝 So, Eq. = = forward-difference in time Spatial derivatives (Eqs. & ) also evaluated at time 𝑝 Substituting all into Eq.→ gives explicit finite-difference equation for interior node ( , ) 𝑚 𝑛
- 50. Discretization of theHeat Equation: The Explicit Method Solving for the nodal temperature at the new (p + 1) time and assuming that ∆x = ∆y, it follows that where Fo is a finite-difference form of the Fourier number If the system is one-dimensional in x, the explicit form of the finite-difference equation for an interior node m reduces to
- 51. Discretization of theHeat Equation: The Explicit Method Equations and are explicit: new temperatures depend only on known previous values. Initial temperatures at all interior nodes are known at t = 0 (p = 0).Calculations start at t = ∆t (p = 1) using Eq. 5.79 or 5.81 for each node. With temperatures at t = ∆t known, apply equations to get values at t = 2∆t (p = 2). Process continues step-by-step in time using time interval ∆t. Transient temperature distribution is built by marching forward in time.
- 52. Discretization of theHeat Equation: The Explicit Method Accuracy improves by reducing ∆x and ∆t. Smaller ∆x → more nodes; smaller ∆t → more time steps. Computation time increases with smaller ∆x and ∆t. ∆x is a trade-off between accuracy and computation. Once ∆x is set, ∆t is chosen for stability, not arbitrarily.
- 53. Discretization of theHeat Equation: The Explicit Method Explicit method is not unconditionally stable. In transient problems, temperatures should approach steady state over time. Explicit method may cause oscillations — not physical, may lead to instability. Instability causes divergence from actual solution. To avoid divergence, ∆t must stay below a certain limit.
- 54. Discretization of theHeat Equation: The Explicit Method Limit depends on ∆x and system parameters — called stability criterion. Criterion derived mathematically or via thermodynamics (e.g., Problem 5.87). For 1D interior node: (1 − 2Fo) ≥ 0 must be satisfied. For prescribed values of ∆x and α, these criteria may be used to determine upper limits to the value of ∆t.
- 55. Discretization of theHeat Equation: The Explicit Method Equations and can be derived using the energy balance method (Section 4.4.3). Apply energy balance to a control volume around an interior node. Include thermal energy storage changes. Leads to a general energy balance equation form. As in Section 4.4.3, we will assume that all heat flow is into the node in the following derivations of the finite difference equations.
- 56. Surface node withconvection and one-dimensional transient conduction.
- 57. Discretization of theHeat Equation: The Explicit Method Use Equation for the surface node in a 1D system (see Figure (Previous slide)). Node spacing is equal; surface node has half the thickness of interior nodes. Assume convection from fluid and no heat generation. Equation simplifies accordingly for surface energy balance.
- 58. Discretization of theHeat Equation: The Explicit Method or, solving for the surface temperature at t + ∆t
- 59. Discretization of theHeat Equation: The Explicit Method To ensure stability, require coefficient of T ≥ 0. ₀ᵖ This condition helps prevent numerical instability. Leads to the stability criterion for the method.
- 60. Discretization of theHeat Equation: The Explicit Method Full solution uses Eq. (interior nodes) and (surface node). Compare Eq. vs. to find stricter stability limit. Since Bi ≥ 0, Fo limit from Eq. < o. Use Eq. to set maximum Fo and ∆t for stability. Table (next slide) lists explicit equations for common geometries. Each derived via energy balance on control volume. For practice, verify at least one equation from the table.
- 61. Discretization of theHeat Equation: The Implicit Method In explicit finite-difference schemes: Temperature at time + 𝑡 Δ𝑡 depends only on known values at time . 𝑡 Each node’s temperature is computed independently of others at + 𝑡 Δ𝑡. Advantage: Simple and computationally convenient. Limitation: Stability requires small Δ𝑡 for a given space step. Often needs many time steps → high computational cost.
- 62. Discretization of theHeat Equation: The Implicit Method Implicit Scheme: Can reduce computation time. Uses backward-difference (Equation ). All temps evaluated at new time ( +1). 𝑝 More stable than explicit method. Suitable for larger Δ . 𝑡 Equation is then considered to provide a backward-difference approximation to the time derivative. In contrast to Equation + , the implicit form of the finite difference equation for the interior node of a two-dimensional system is then
- 63. Discretization of theHeat Equation: The Implicit Method In contrast to Equation + , the implicit form of the finite difference equation for the interior node of a two-dimensional system is then
- 64. Discretization of theHeat Equation: The Implicit Method (B)
- 65. Discretization of theHeat Equation: The Implicit Method Eq. B: New temperature at (m, n) depends on adjoining nodes' new temperatures. These neighboring node temperatures are generally unknown. Must solve all nodal equations simultaneously for time step +1. 𝑝 Use matrix inversion or Gauss–Seidel iteration (see Sec. 4.5 i.e., Solving the Finite-Difference Equations, App. D). Marching solution: Solve at = 𝑡 Δ ,2Δ ,… 𝑡 𝑡 … until final time.
- 66. Discretization of theHeat Equation: The Implicit Method Implicit method is unconditionally stable. All coefficients of are positive. No restrictions on Δ𝑥 and Δ𝑡. Larger Δ𝑡 → faster computation, minimal accuracy loss. To ensure accuracy, use small enough Δ𝑡 so results don’t change with further reductions.
- 67. Discretization of theHeat Equation: The Implicit Method The implicit form of a finite-difference equation may also be derived from the energy balance method. For the surface node of Figure , it is readily shown that
- 68. Discretization of theHeat Equation: The Implicit Method For any interior node of Figure, it may also be shown that