Time-Dependent Conduction :

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1 Time-Dependent Conduction : The Lumped Capacitance Method Chapter Five Sections 5.1 thru 5.3

2 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 and proceeds until a new steady state (thermal equilibrium) is achieved. It can be induced by changes in: surface convection conditions ( h, T ), h, T surface radiation conditions ( ), Solution Techniques The Lumped Capacitance Method Exact Solutions The Finite-Difference Method r sur a surface temperature or heat flux, and/or internal energy generation.

5.1 The Lumped Capacitance Method Based on the assumption of a spatially uniform temperature distribution throughout the transient process. Hence,.

3 5.1 The Lumped Capacitance Method Based on the assumption of a spatially uniform temperature distribution throughout the transient process. Hence,. T( r, t) Why is the assumption never fully realized in practice? () T t General Lumped Capacitance Analysis: Consider a general case, which includes convection, radiation and/or an applied heat flux at specified surfaces ( As, c, As, r, As, h), as well as internal energy generation.

4 First Law: de dt st dt i i i = ρ c = E E + E dt in out g Assuming energy outflow due to convection and radiation and with inflow due to an applied heat flux q s, dt ρ c = q A ha ( T T ) h A ( T T ) + E i dt s s, h s, c r s, r sur g Is this expression applicable in situations for which convection and/or radiation provide for energy inflow? May h and h r be assumed to be constant throughout the transient process?

5 ( ) Special Cases (Exact Solutions, T T ) 0 i Negligible Radiation ( θ T T θ θ b a), / : i a hasc, / ρ c b q sash, E + g / ρ c The non-homogeneous differential equation is transformed into a homogeneous equation of the form: dθ dt = aθ

6 dθ = adt θ Integrating from t=0 to any t and rearranging, ln( θ ) = at + C θ = exp( at + C ) 1 1 θ = C exp( at) or θ b/ a = C exp( at) 2 2 at t = 0, T = T C = ( T T ) b/ a i Hence, ( T T ) b/ a = [( T T ) b/ a] exp( at) T T b/ a b/ a = + [1 ] exp( at) T T T T T T i i i i 2 i T T b/ a or, = exp ( at) + 1 exp( at) T T T T (5.25) i i To what does the foregoing equation reduce as steady state is approached? How else may the steady-state solution be obtained?

7 Negligible Radiation and Source Terms (h >> h, E g = 0, q = 0): dt ρ c = hasc, ( T T ) (5.2) dt r i s ρ c ha θ dθ θ = 0 sc, θi t dt θ T T ha sc, t = = exp t = exp θi Ti T ρ c τt

8 The thermal time constant is defined as τ t 1 ha sc, ( ρ c) Thermal Resistance, R t Lumped Thermal Capacitance, C t (5.7) The change in thermal energy storage due to the transient process is t Est Q = E i t t outdt = hasc, θ dt = ( ρ c) θi 1 exp 0 0 τt (5.8)

9 Negligible Convection and Source Terms h >> h, E g = 0, q = 0 : Assuming radiation exchange with large surroundings, dt ρ c = εasr, σ T T dt 4 4 ( ) sur r i s ε A sr, σ ρ c t dt = T 0 T i T dt T 4 4 sur t ρ c Tsur + T Tsur + Ti = 1n 1n 4εA 3 sr, σtsur Tsur T Tsur Ti (5.18) T T 2 tan 1 tan 1 i + Tsur Tsur Result necessitates implicit evaluation of T(t).

10 5.2 Validity of The Lumped Capacitance Method The Biot Number: The first of many dimensionless parameters to be considered. Definition: Bi Biot number = hl / k c h convection or radiation coefficient k thermal conductivity of the solid L c characteristic length of the solid ( / A or coordinate associated with maximum spatial temperature difference) s Physical Interpretation: Bi = L / ka R T 1/ ha R T c s cond solid s conv solid / fluid Criterion for Applicability of Lumped Capacitance Method: Bi << 1

11 ka : ( Ts,1 Ts,2 ) = ha( Ts,2 T ) L Tsi, Ts,2 L/ kas Rcond hl Physical Interpretation: = = = = T T 1/ ha R k s,2 s conv hlc Bi < 0.1 ( Lumped Heat Capacitance ) k L = V / A c Plate : L = L; s c Cylinder : L = r / 2 ; Sphere : L = r /3 ( L = r ) c o c o c o Bi

12 : T T hasc, hl A k c sc, (5.6) = exp t = exp ( )( ) t T T ρ c k ρ cl i c hlc k/ ρc hlc αt = exp ( )( ) t exp ( )( ) exp ( Bi)( Fo ) 2 = 2 k L k L c c αt Fo = Fourier number L 2 c [ ]

23 Problem 5.12: Charging a thermal energy storage system consisting of a packed bed of aluminum spheres. KNOWN: Diameter, density, specific heat and thermal conductivity of aluminum spheres used in packed bed thermal energy storage system. Convection coefficient and inlet gas temperature. FIND: Time required for sphere at inlet to acquire 90% of maximum possible thermal energy and the corresponding center temperature. Schematic:

24 ASSUMPTIONS: (1) Negligible heat transfer to or from a sphere by radiation or conduction due to contact with other spheres, (2) Constant properties. ANALYSIS: To determine whether a lumped capacitance analysis can be used, first compute Bi = h(r o /3)/k = 75 W/m 2 K (0.025m)/150 W/m K = <<1. Hence, the lumped capacitance approximation may be made, and a uniform temperature may be assumed to exist in the sphere at any time. From Eq. 5.8a, achievement of 90% of the maximum possible thermal energy storage corresponds to E st = 0.90 = 1 exp t / ρcvθi t ( ) ( τ ) t = τt ln 0.1 = 427s 2.30 = 984s t kg / m 0.075m 950 J / kg K = Vc / has = Dc / 6h = = 427s W/m K τ ρ ρ

25 From Eq. (5.6), the corresponding temperature at any location in the sphere is ( ) T 984s ( 2 3 ) = 300 C 275 C exp 6 75 W / m K 984s / 2700 kg / m 0.075m 950 J / kg K ( ) = + ( ) ( ρ ) T 984s Tg,i Ti Tg,i exp 6ht / Dc T( 984s) = C If the product of the density and specific heat of copper is (ρc) Cu 8900 kg/m J/kg K = J/m 3 K, is there any advantage to using copper spheres of equivalent diameter in lieu of aluminum spheres? Does the time required for a sphere to reach a prescribed state of thermal energy storage change with increasing distance from the bed inlet? If so, how and why?

QUESTION ANSWER. . e. Fourier number:

QUESTION ANSWER. . e. Fourier number: QUESTION 1. (0 pts) The Lumped Capacitance Method (a) List and describe the implications of the two major assumptions of the lumped capacitance method. (6 pts) (b) Define the Biot number by equations and