Transient Conduction: The Lumped Capacitance Method Chapter Five Sections 5.1 through 5.3

Transient Conduction Transient Conduction A heat transfer process for which the temperature varies with time, as well as location within a solid. It is initiated whenever a system experiences a change in operating conditions. It can be induced by changes in: surface convection conditions ( ), surface radiation conditions ( ), a surface temperature or heat flux, and/or internal energy generation. Solution Techniques The Lumped Capacitance Method Exact Solutions The Finite-Difference Method

Lumped Capacitance Method The Lumped Capacitance Method Based on the assumption of a spatially uniform temperature distribution throughout the transient process. Hence, . Why is the assumption never fully realized in practice? General Lumped Capacitance Analysis: Consider a general case, which includes convection, radiation and/or an applied heat flux at specified surfaces as well as internal energy generation

Lumped Capacitance Method (cont.) First Law: Assuming energy outflow due to convection and radiation and inflow due to an applied heat flux (5.15) May h and hr be assumed to be constant throughout the transient process? How must such an equation be solved?

Special Case (Negligible Radiation) Special Cases (Exact Solutions, ) Negligible Radiation The non-homogeneous differential equation is transformed into a homogeneous equation of the form: Integrating from t = 0 to any t and rearranging, (5.25) To what does the foregoing equation reduce as steady state is approached? How else may the steady-state solution be obtained?

Special Case (Convection) Negligible Radiation and Source Terms (5.2) Note: (5.6) The thermal time constant is defined as (5.7) Thermal Resistance, Rt Lumped Thermal Capacitance, Ct The change in thermal energy storage due to the transient process is (5.8)

Special Case (Radiation) Negligible Convection and Source Terms Assuming radiation exchange with large surroundings, (5.18) This result necessitates implicit evaluation of T(t).

The Biot Number and Validity of The Lumped Capacitance Method The Biot Number: The first of many dimensionless parameters to be considered. Definition: Physical Interpretation: See Fig. 5.4. Criterion for Applicability of Lumped Capacitance Method:

Problem: Thermal Energy Storage Problem 5.12: Charging a thermal energy storage system consisting of a packed bed of aluminum spheres. Schematic:

Problem: Thermal Energy Storage (cont.) ANALYSIS: To determine whether a lumped capacitance analysis can be used, first compute Bi = h(ro/3) = 75 W/m2∙K (0.0125 m)/150 W/m∙K = 0.006 << 1. From Eq. 5.8, achievement of 90% of the maximum possible thermal energy storage corresponds to < <

Problem: Furnace Start-up Problem 5.22: Heating of coated furnace wall during start-up. Schematic:

Problem: Furnace Start-up (cont.) Hence, with <

Problem: Furnace Start-up (cont.) <

Chapter 5 Sections 5.4 through 5.8 Transient Conduction: Spatial Effects and the Role of Analytical Solutions Chapter 5 Sections 5.4 through 5.8

Solution to the Heat Equation for a Plane Wall with Symmetrical Convection Conditions If the lumped capacitance approximation cannot be made, consideration must be given to spatial, as well as temporal, variations in temperature during the transient process. For a plane wall with symmetrical convection conditions and constant properties, the heat equation and initial/boundary conditions are: (5.29) (5.30) (5.31) (5.32) Existence of eight independent variables: (5.33) How may the functional dependence be simplified?

Dimensionless temperature difference: Plane Wall (cont.) Non-dimensionalization of Heat Equation and Initial/Boundary Conditions: Dimensionless temperature difference: Dimensionless space coordinate: Dimensionless time: The Biot Number: Exact Solution: (5.42a) (5.42b,c) See Appendix B.3 for first four roots (eigenvalues ) of Eq. (5.42c).

The One-Term Approximation : Plane Wall (cont.) The One-Term Approximation : Variation of midplane temperature (x*= 0) with time : (5.44) Variation of temperature with location (x*) and time : (5.43b) Change in thermal energy storage with time: (5.46a) (5.49) (5.47) Can the foregoing results be used for a plane wall that is well insulated on one side and convectively heated or cooled on the other? Can the foregoing results be used if an isothermal condition is instantaneously imposed on both surfaces of a plane wall or on one surface of a wall whose other surface is well insulated?

Graphical Representation of the One-Term Approximation Heisler Charts Graphical Representation of the One-Term Approximation The Heisler Charts, Section 5 S.1 Midplane Temperature:

Temperature Distribution: Heisler Charts (cont.) Temperature Distribution: Change in Thermal Energy Storage:

Radial Systems Long Rods or Spheres Heated or Cooled by Convection. One-Term Approximations: Long Rod: Eqs. (5.52) and (5.54) Sphere: Eqs. (5.53) and (5.55) Graphical Representations: Long Rod: Figs. 5 S.4 – 5 S.6 Sphere: Figs. 5 S.7 – 5 S.9

The Semi-Infinite Solid A solid that is initially of uniform temperature Ti and is assumed to extend to infinity from a surface at which thermal conditions are altered. Special Cases: Case 1: Change in Surface Temperature (Ts) (5.60) (5.61)

Semi-Infinite Solid (cont.) Case 2: Constant Heat Flux (5.62) Case 3: Surface Convection (5.63)

Multidimensional Effects Solutions for multidimensional transient conduction can often be expressed as a product of related one-dimensional solutions for a plane wall, P(x,t), an infinite cylinder, C(r,t), and/or a semi-infinite solid, S(x,t). See Equations (5 S.1) to (5 S.3) and Fig. 5 S.11. Consider superposition of solutions for two-dimensional conduction in a short cylinder:

Objects with Constant Ts or qs Objects with Constant Surface Temperatures or Surface Heat Fluxes Transient response of a variety of objects to a step change in surface temperature or heat flux can be unified by defining the dimensionless conduction heat rate: (5.67) where Lc is a characteristic length that depends on the geometry of the object. Consider the variation of q* with time, or Fo, for Interior heat transfer: Heat transfer inside objects such as plane walls, cylinders, or spheres, Exterior heat transfer: Heat transfer in an infinite medium surrounding an embedded object.

Objects with Constant Ts or qs (cont.) When q* is plotted versus Fo in Figure 5.10, we see that: All objects behave the same as a semi-infinite solid for short times. q* approaches a steady state for exterior objects. q* does not reach a steady state for interior objects, but decreases continually with time (Fo). Constant Ts Constant qs Why do all objects behave the same as a semi-infinite solid for short times?

Objects with Constant Ts or qs (cont.) Approximate Solutions for Objects with Constant Ts or qs Easy-to-use approximate solutions for q*(Fo) are presented in Table 5.2 for all the cases presented in Figure 5.10. As an example of the use of Table 5.2, consider: Infinite cylinder initially at Ti has constant heat flux imposed at its surface. Find its surface temperature as a function of time. Look in Table 5.2b for constant surface heat flux, Interior Cases, Infinite cylinder. Length scale is Lc = ro, the cylinder radius. Exact solution for q*(Fo) is a complicated infinite series. Approximate solution is given by: It is then a simple matter to find Ts from the definition,

Problem: Thermal Energy Storage Problem 5.80: Charging a thermal energy storage system consisting of a packed bed of Pyrex spheres.

Problem: Thermal Energy Storage (cont.)

Problem: Thermal Energy Storage (cont.) < < <

Problem: Thermal Response Firewall Problem: 5.93: Use of radiation heat transfer from high intensity lamps for a prescribed duration (t=30 min) to assess ability of firewall to meet safety standards corresponding to maximum allowable temperatures at the heated (front) and unheated (back) surfaces.

Problem: Thermal Response of Firewall (cont.)

Problem: Thermal Response of Firewall (cont.) <

Problem: Microwave Heating Problem: 5.101: Microwave heating of a spherical piece of frozen ground beef using microwave-absorbing packaging material.

Problem: Microwave Heating (cont.) Thus The beef can be seen as the interior of a sphere with a constant heat flux at its surface, thus the relationship in Table 5.2b, Interior Cases, sphere, can be used. We begin by calculating q* for Ts=0°C.

Problem: Microwave Heating (cont.) Since this is less than 0.2, our assumption was correct. Finally we can solve for the time: < COMMENTS: At the minimum surface temperature of -20°C, with T∞ = 30°C and h = 15 W/m2∙K from Problem 5.33, the convection heat flux is 750 W/m2, which is less than 8% of the microwave heat flux. The radiation heat flux would likely be less, depending on the temperature of the oven walls.

Transient Conduction: Finite-Difference Equations and Solutions Chapter 5 Section 5.10

Finite-Difference Method The Finite-Difference Method An approximate method for determining temperatures at discrete (nodal) points of the physical system and at discrete times during the transient process. Procedure: Represent the physical system by a nodal network, with an m, n notation used to designate the location of discrete points in the network, and discretize the problem in time by designating a time increment ∆t and expressing the time as t = p∆t, where p assumes integer values, (p = 0, 1, 2,…). Use the energy balance method to obtain a finite-difference equation for each node of unknown temperature. Solve the resulting set of equations for the nodal temperatures at t = ∆t, 2∆t, 3∆t, ... What is represented by the temperature, ?

Energy Balance and Finite-Difference Energy Storage Term Energy Balance and Finite-Difference Approximation for the Storage Term For any nodal region, the energy balance is (5.84) where, according to convention, all heat flow is assumed to be into the region. Discretization of temperature variation with time: (5.77) Finite-difference form of the storage term: Existence of two options for the time at which all other terms in the energy balance are evaluated: p or p+1.

The Explicit Method of Solution All other terms in the energy balance are evaluated at the preceding time corresponding to p. Equation (5.77) is then termed a forward-difference approximation. Example: Two-dimensional conduction for an interior node with ∆x=∆y. (5.79) Unknown nodal temperatures at the new time, t = (p+1)∆t, are determined explicitly by known nodal temperatures at the preceding time, t = p∆t, hence the term explicit solution. We solve by marching in time.

Explicit Method (cont.) How is solution accuracy affected by the choice of ∆x and ∆t? Do other factors influence the choice of ∆t? What is the nature of an unstable solution? Stability criterion: Determined by requiring the coefficient for the node of interest at the previous time to be greater than or equal to zero. For a finite-difference equation of the form, Hence, for the two-dimensional interior node: or Therefore Table 5.3 finite-difference equations for other common nodal regions. Very small time steps may be required for large numbers of nodes (small Dx).

The Implicit Method of Solution Addresses the limitations of the explicit method, allowing larger time steps. All other terms in the energy balance are evaluated at the new time corresponding to p+1. Equation (5.77) is then termed a backward-difference approximation. Example: Two-dimensional conduction for an interior node with ∆x=∆y. (5.95) System of N finite-difference equations for N unknown nodal temperatures may be solved by matrix inversion or Gauss-Seidel iteration. Solution is unconditionally stable.

Marching Solution Marching Solution Transient temperature distribution is determined by a marching solution, beginning with known initial conditions. Known p t T1 T2 T3……………….. TN 0 0 T1,i T2,i T3,i………………. TN,i 1 ∆t -- -- -- …………… -- 2 2∆t -- -- -- …………… -- 3∆t -- -- -- …………… -- . . Steady state -- -- -- -- …………… --

Problem: Finite-Difference Equation Problem 5.109: Derivation of explicit form of finite-difference equation for a nodal point in a thin, electrically conducting rod confined by a vacuum enclosure. SCHEMATIC:

Problem: Finite-Difference Equation (cont.)

Problem: Finite-Difference Equation (cont.) < However, stability is also affected by the nonlinear term, , and smaller values of Fo may be needed to insure a stable solution.

Problem: Implicit Finite-Difference Method Problem 5.121: Use of implicit finite-difference method to determine transient response of an acrylic sheet suddenly brought into contact with a hot steel plate. KNOWN: Dimensions and properties of acrylic (A) and steel (B) plates. Initial temperatures. FIND: Using 20 equally-spaced nodes, find time needed to bring external surface of acrylic to Tsoft = 90°C. Plot the spatially-averaged acrylic and steel plate temperatures for 0 ≤ t ≤ 300 s. Schematic : LA = LB = 5 mm Ti,A = 20°C Ti,B = 300°C

Problem: Softening Acrylic (cont.) PROPERTIES: Acrylic (given): rA = 1990 kg/m3, cA = 1470 J/kgK and kA = 0.21 W/mK. Steel (given): rB = 7800 kg/m3, cB = 500 J/kgK and kB = 45 W/mK. ANALYSIS: We begin by writing energy balances on each of the 20 control volumes using the implicit method, Node 1: Nodes 2 – 9: Node 10:

Problem: Softening Acrylic (cont.) Node 11: Nodes 12 – 19: Node 20: where

Problem: Softening Acrylic (cont.) The average temperature in each material may be expressed as or, in finite difference form and

Problem: Softening Acrylic (cont.) The preceding equations were solved using the implicit method. The average temperatures of the acrylic and steel, as well as the back side (insulated) surface temperature of the acrylic (T1) are shown below. < From the simulation, we also find that the surface of the acrylic reaches Tsoft = 90C at t = 87 s. < COMMENTS: (1) Ultimately, the temperatures of the two plates will reach the same value. The steady-state temperature may be found by recognizing the energy gained by the acrylic is lost by the steel, LAcA(Tss – Ti,A) = LBcB(Ti,B – Tss) yielding Tss = 180C. (2) Can you explain the behavior shown in the graph?