Análise do Método Capacitivo em Condução Transiente

Condução Transiente Método Capacitivo Cap 5 Condução Transiente Alterações na condição de equilíbrio térmico convecção na superfície h T Técnicas de solução Método Capacitivo TTt Soluções Exatas TTxyzt Métodos numéricos radiação na superfície r sur h T temperatura ou fluxo de calor imposto na superfície geração interna de calor T r t T t Método Capacitivo Ti TTi Tt t 0 h T r t T t Txyzt1 Tt1 Ttn Método Capacitivo Ti TTi Tt t 0 VC h Balanço de Energia dt mc dT T T t hA E dt dE E E E s ac ar s g e s c c dT hA T T dt 52 0 i s c t c d hA dt s c i i hA T T exp t T T c t t exp dt mc dT T T t hA E dt dE E E E s ac ar s g e 1 t s c c hA 57 Resistência térmica Rt Capacitância térmica Ct Calor trocado durante o processo 0 t out Est Q E dt 0 t s c hA dt 1 exp i t t c 58 A constante de tempo térmica é definida como Análise do método Capacitivo para uma parede plana Balanço de Energia st in out g dE c dT E E E dt dt g s s h s c r s r sur c dT q A hA T T h A T T E dt Casos especiais Sol Exata 0 i T T Sem radiação T T b a g s c s s h a hA c b q A E c d a dt exp 1 exp i i T T b a at at T T T T 525 Sem radiação e sem geração 0 0 g r s h h E q s c c dT hA T T dt 52 0 i s c t c d hA dt s c i i hA T T exp t T T c t t exp Sem convecção e sem geração 0 0 g r s h h E q 4 4 s r sur c dT A T T dt 4 4 0 i s r T sur T t A c dT T T dt 3 1n 1n 4 sur sur i s r sur sur sur i T T T T c t A T T T T T 518 1 1 2 tan tan i sur sur T T T T Número de Biot e validade do método Capacitivo O Número de Biot análise adimensional Definição hLc Bi k Interpretação física S C A V L LcComprimento característico conv cond S S C S S S S S S conv cond R R hA kA L T T T T i T hA T T L T kA q q 1 2 2 1 2 2 1 Critério de aplicação do método capacitivo Critério de aplicação do método capacitivo 1 Bi Admitese Bi01 Comprimento Característico S C A V L Para uma parede de espessura 2L LcL Para um cilindro de raio R LcR2 Para uma esfera de raio R LcR3 Análise Adimensional Vct hA i s e Fo i L t k hL L t c k k hL cL ht Vc t hA C C C C C s 2 2 Fo i i e Problem Thermal Energy Storage Problem 512 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 Aluminum sphere D 75 mm T 25 C i o Gas T C gi o 300 h 75 Wm K 2 2700 kgm 3 k 240 WmK c 950 JkgK Schematic Problem Thermal Energy Storage cont 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 hro3k 75 Wm2K 0025m150 WmK 0013 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 58a achievement of 90 of the maximum possible thermal energy storage corresponds to st t i E 090 1 exp t cV t t ln 01 427s 230 984s From Eq 56 the corresponding temperature at any location in the sphere is gi i gi T 984s T T T exp 6ht Dc 2 3 T 984s 300 C 275 C exp 6 75 W m K 984s 2700 kg m 0075m 950 J kg K If the product of the density and specific heat of copper is cCu 8900 kgm3 400 JkgK 356 106 Jm3K 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 T 984s 2725 C 3 t s 2 2700 kg m 0075m 950 J kg K Vc hA Dc 6h 427s 6 75 W m K Problem Furnace Startup Problem 516 Heating of coated furnace wall during startup KNOWN Thickness and properties of furnace wall Thermal resistance of ceramic coating on surface of wall exposed to furnace gases Initial wall temperature FIND a Time required for surface of wall to reach a prescribed temperature b Corresponding value of coating surface temperature Schematic Problem Furnace Startup ASSUMPTIONS 1 Constant properties 2 Negligible coating thermal capacitance 3 Negligible radiation PROPERTIES Carbon steel 7850 kgm3 c 430 JkgK k 60 WmK ANALYSIS Heat transfer to the wall is determined by the total resistance to heat transfer from the gas to the surface of the steel and not simply by the convection resistance 1 1 1 2 2 2 tot f 2 1 1 U R R 10 m KW 20 Wm K h 25 Wm K 2 UL 20 Wm K 001 m Bi 00033 1 k 60 Wm K and the lumped capacitance method can be used a From Eqs 56 and 57 t t t i T T exp t exp tR C exp Ut Lc T T 3 2 i 7850 kgm 001 m 430 Jkg K T T Lc 1200 1300 t ln ln U T T 300 1300 20 Wm K t 3886s 108h Hence with Problem Furnace Startup cont b Performing an energy balance at the outer surface so so so si f h T T T T R 2 2 2 si f so 2 f hT T R 25 Wm K 1300 K 1200 K10 m KW T h 1 R 25 100 Wm K Tso 1220 K How does the coating affect the thermal time constant Condução transiente consideração dos efeitos espaciais e as soluções analíticas Soluções analíticas para problemas transientes quando o método da capacitância global não é válido Bi não é 1 Para uma placa placa com propriedades constantes e para as seguintes condições de contorno 2 2 1 T T x t 526 0 i T x T 527 0 0 x T x 528 x L k T h T L t T x 529 i T T x t T T k h 530 TTxyzt Solução adimensional Diferença de Temp adimensional i i T T T T x x L Coordenadas adim Tempo adim 2 t t Fo L Número de Biot solid hL Bi k f x Fo Bi Solução Exata 2 1 exp cos n n n n C Fo x 539a 4sin tan 2 sin 2 n n n n n n C Bi 539bc Apendice B3 para as raízes eigenvalues 1 4 Coefficients used in the oneterm approximation to the series solutions for transient onedimensional conduction Representação Gráfica para parede plana Heisler Cartas Temperatura média do meio função do tempo método capacitivo Distribuição de Temp Variações no calor transferido Sistemas Radiais 2 o o Bi hr k Fo t r OneTerm Approximations Long Rod Eqs 549 and 551 Sphere Eqs 550 and 552 1 1 Table 51 C Graphical Representations Long Rod Figs 5 S4 5 S6 Sphere Figs 5 S7 5 S9 Sólido SemiInfinito Caso 1 mudança na Temp de superfície Ts 0 0 s i T t T T x T x erf 2 t s i s T x t T T T 557 s i s k T T q t 558 12 2 2 exp 4 erfc 2 o i o q t x T x t T k t q x x k t 559 Caso 2 Fluxo de calor cte s o q q 0 0 x k T h T T t x 2 2 2 2 i i T x t T x erfc T T t hx h t x h t exp erfc k k k t 560 Caso 3 Convecção na sup h T Transient temperature distributions in a semiinfinite solid for three surface conditions constant surface temperature constant surface heat flux and surface convection Problemas com interface de contato Aquecimento periódico na superfície Summary of transient heat transfer results for constant surface temperature cases Summary of transient heat transfer results for constant surface heat flux cases Efeitos Multidimensionais Superposição de soluções método de separação de vairiáveis i Plane Infinite i i Wall Cylinder T r x t T P x t C r t T T T x t T T rt T T T T T Transient Conduction FiniteDifference Equations and Solutions Não será cobrado FiniteDifference Method The FiniteDifference 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 Use the energy balance method to obtain a finitedifference equation for each node of unknown temperature Solve the resulting set of equations for the nodal temperatures at t t 2t 3t until steadystate is reached What is represented by the temperature p m n T and discretize the problem in time by designating a time increment t and expressing the time as t pt where p assumes integer values p 0 1 2 Storage Term Energy Balance and FiniteDifference Approximation for the Storage Term For any nodal region the energy balance is in g st E E E 581 where according to convention all heat flow is assumed to be into the region Discretization of temperature variation with time Finitedifference form of the storage term 1 p p m n m n st m n T T E c t Existence of two options for the time at which all other terms in the energy balance are evaluated p or p1 1 p p m n m n m n T T T t t 574 Explicit Method The Explicit Method of Solution All other terms in the energy balance are evaluated at the preceding time corresponding to p Equation 574 is then termed a forwarddifference approximation Example Twodimensional conduction for an interior node with xy 1 1 1 1 1 1 4 p p p p p p m n m n m n m n m n m n T Fo T T T T Fo T 576 2 finitedifference form o Four ier f number t Fo x Unknown nodal temperatures at the new time t p1t are determined exclusively by known nodal temperatures at the preceding time t pt hence the term explicit solution 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 1 0 p p m n m n T AT A Hence for the twodimensional interior node 1 4 0 Fo 14 Fo 2 4 x t Table 53 finitedifference equations for other common nodal regions For a finitedifference equation of the form Implicit Method The Implicit Method of Solution All other terms in the energy balance are evaluated at the new time corresponding to p1 Equation 574 is then termed a backwarddifference approximation Example Twodimensional conduction for an interior node with xy 1 1 1 1 1 1 1 1 1 1 4 p p p p p p m n m n m n m n m n m n Fo T Fo T T T T T 592 System of N finitedifference equations for N unknown nodal temperatures may be solved by matrix inversion or GaussSeidel iteration Solution is unconditionally stable Table 53 finitedifference equations for other common nodal regions Marching Solution Marching Solution Transient temperature distribution is determined by a marching solution beginning with known initial conditions 1 t 2 2t 3 3t Steadystate Known p t T1 T2 T3 TN 0 0 T1i T2i T3i TNi Problem FiniteDifference Equation Problem 593 Derivation of explicit form of finitedifference equation for a nodal point in a thin electrically conducting rod confined by a vacuum enclosure KNOWN Thin rod of diameter D initially in equilibrium with its surroundings Tsur suddenly passes a current I rod is in vacuum enclosure and has prescribed electrical resistivity e and other thermophysical properties FIND Transient finitedifference equation for node m SCHEMATIC Problem FiniteDifference Equation ASSUMPTIONS 1 Onedimensional transient conduction in rod 2 Surroundings are much larger than rod 3 Constant properties ANALYSIS Applying conservation of energy to a nodal region of volume Ac x where 2 Ac D 4 in out g st E E E E p1 p 2 m m a b rad e T T q q q I R cV t Hence with 2 g e E I R where e e c R xA and use of the forwarddifference representation for the time derivative 4 p p p p p1 p 4 m m p 4 2 e m1 m1 m m c c m sur c c T T T T x T T kA kA D x T T I cA x x x A t Dividing each term by cAc xt and solving for p1 Tm p1 p p p m m m1 m1 2 2 k t k t T T T 2 1 T c c x x 2 p 4 4 e m sur 2 c c I P t t T T A c c A Problem FiniteDifference Equation or with Fo tx2 2 2 2 4 p1 p p p p 4 e m m m sur m1 m1 2 c c I x P x T Fo T T 1 2 Fo T Fo T T Fo kA kA Basing the stability criterion on the coefficient of the p Tm term it would follow that Fo ½ However stability is also affected by the nonlinear term p 4 Tm and smaller values of Fo may be needed to insure its existence