Time-Dependent Conduction : The Lumped Capacitance Method Chapter Five Sections 5.1 thru 5.3
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,. 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.
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?
( ) 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
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?
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
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)
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).
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
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
: 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 [ ]
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:
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 = 0.013 <<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 3 2700 kg / m 0.075m 950 J / kg K = Vc / has = Dc / 6h = = 427s. 2 6 75W/m K τ ρ ρ
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) = 272.5 C If the product of the density and specific heat of copper is (ρc) Cu 8900 kg/m 3 400 J/kg K = 3.56 10 6 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?