Método Integral Placa plana Escoamento paralelo

Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa Os principais parâmetros da convec ção podem ser obtidos através de métodos de resolução das equações da camada limite. Para ilustrar, iremos supor o seguinte caso para o escoamento paralelo em uma placa plana: \u2022 Escoamento laminar \u2022 Incompressível \u2022 Regime permanente \u2022 Propriedades do fluido constantes \u2022 Dissipação viscosa desprezível \u2022 Ausência de queda de pressão \uf23a\ue740\ue732 \ue200\ue740\ue754 \ued4c \ue372\uf23b Nossas equações fundamentais (Continuidade, Movimento e Energ ia) são: \u2022 Continuidade: \ue7f2\ue7e9 \ue7f2\ue750 \ued45 \ue7f2 \ue7f2\ue754 \uf23a \ue7e9\ue752 \uebeb \uf23b \ued45 \ue7f2 \ue7f2\ue755 \ued6b\ue7e9\ue752 \uebec \ued6f \ued45 \ue7f2 \ue7f2\ue756 \uf23a \ue7e9\ue752 \uebed \uf23b \ued4c \ue372 \u2022 Movimento (direção x): \ue7e9 \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue750 \ued45 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued45 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \ued45 \ue752 \uebed \ue7f2 \ue752 \uebeb \ue7f2\ue756 \ued70 \ued4c \ued46 \ue7f2\ue732 \ue7f2\ue754 \ued45 \ue7e4 \uf248 \ue7f2\ue37e \ue752 \uebeb \ue7f2\ue754 \ue37e \ued45 \ue7f2\ue37e \ue752 \uebeb \ue7f2\ue755 \ue37e \ued45 \ue7f2\ue37e\ue752 \uebeb \ue7f2\ue756 \ue37e \uf249 \ued45 \ue7e9\ue743 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued45 \ue7f2 \ue752 \uebec \ue7f2\ue755 \ued4c \ue372 Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa Supondo que \ue752 \uebeb \ued4c \ue742 \uf23a\ue754 \ue1e1 \ue755 \uf23b além do fato do componente da gravidade na direção \ue754 ser nulo: \ue7e9 \ued6c\ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued45 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \ued70 \ued4c \ue7e4 \uf246 \ue7f2 \ue37e\ue752 \uebeb \ue7f2\ue754 \ue37e \ued45 \ue7f2\ue37e \ue752 \uebeb \ue7f2\ue755 \ue37e \uf247 Também supondo que a dependência de \ue752 \uebeb é muito maior em relação ao eix o \ue755 do que ao eixo \ue754 . Em outras palavr as L >>>> \ue7dc ; ou seja, comprimento da pla ca muito maior que a espessura da c amada limite. I sso faz com que os gradientes normais à superf ície do objeto sejam muito maiores do que aqueles ao longo da superfície. Isso ficam mais claro observando a figura abaixo: O gradiente na direção y é dado por: \ue752 \uebeb \ueb35 \ued46 \ue752 \uebeb \ueb34 \ue72e Já o gradiente na direção x é dado por: \ue752 \uebeb \ueb36 \ued46 \ue752 \uebeb \ueb34 \ue72e Como a espessura da camada limite de velocidade é pequena, temos um grande gradiente entre a ve locidade próxima à placa, \ue752 \ueb34 , e a ve locidade próxima à corrente livre , \ue752 \ueb35 , pois a placa está estagnada (exemplo da figura) ou com velocidade d iferente da corrente livre. Já na direção \ue754 a variação na velocidade não é sentida tão forte mente como na direção \ue755 , fazendo com que \ue752 \uebeb \ueb35 \ued46 \ue752 \uebeb \ueb34 \ue72e \ue628 \ue752 \uebeb \ueb36 \ued46 \ue752 \uebeb \ueb34 \ue72e Sendo assim, podemos dizer que: Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa \ue7f2 \ue37e\ue752 \uebeb \ue7f2\ue755 \ue37e \ue628\ue628 \ue7f2\ue37e \ue752 \uebeb \ue7f2\ue754 \ue37e Pode- se então desprezar a derivada segunda em r elação a x. Além diss o, podemos acrescente a definição de viscosidade cinemática, que é dada por \ue253 \ued4c \ue7e4 \ue7e9 \ue3a4 \ue003 . Então: \u2022 Energia \ue7e9\ue725 \uf218 \uebe3 \ued6c \ue7f2\ue736 \ue7f2\ue750 \ued45 \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ued45 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ued45 \ue752 \uebed \ue7f2\ue736 \ue7f2\ue756 \ued70 \ued4c \ue747 \uf248 \ue7f2\ue37e\ue736 \ue7f2\ue754 \ue37e \ued45 \ue7f2\ue37e\ue736 \ue7f2\ue755 \ue37e \ued45 \ue7f2\ue37e\ue736 \ue7f2\ue756 \ue37e \uf249 \ued45 \ue003\ue7e4 \ue230 \ueb74 Relembrando que supo mos que a perda por dissi pação viscos a é desprezível, e considerando que a esp essura da camada limi te d e temperatura é mui to menor do que o comprimento da placa: \ue7e4 \ue230 \ueb74 \ued4c \ue372 E \ue7f2\ue37e\ue736 \ue7f2\ue755 \ue37e \ue628 \ue7f2\ue37e\ue736 \ue7f2\ue754 \ue37e Sendo também \ue736 \ued4c \ue742\uf23a \ue754 \ue1e1 \ue755\uf23b Temos então: \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ued45 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ued4c \ue747 \ue7e9\ue725 \uf218 \uebe3 \ue7f2\ue37e\ue736 \ue7f2\ue755 \ue37e Definindo o coeficiente de condutividade térmica \ue7d9 \ued4c \uebde \uec18\uebbc \uf218 \uecdb \ue003 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued45 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \ued4c \ueacd \ue7f2 \ue37e\ue752 \uebeb \ue7f2\ue755 \ue37e \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ued45 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ued4c \ue7d9 \ue7f2\ue37e\ue736 \ue7f2\ue755 \ue37e Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa Em resumo: \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued45 \ue7f2 \ue752 \uebec \ue7f2\ue755 \ued4c \ue372 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued45 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \ued4c \ueacd \ue7f2 \ue37e\ue752 \uebeb \ue7f2\ue755 \ue37e \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ued45 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ued4c \ue8bb \ue7f2\ue37e\ue736 \ue7f2\ue755 \ue37e MÉTODO INTEGRAL DE RESOLUÇÃO DAS EQUAÇÕES DA CAMADA LIMITE Uma abordage m alternativa para a solução das equações da camada-limite envolve o uso de um método integral aproximado . Essa abordagem não possui complicações matemáticas do método exato (similaridade), porém ela pode ser utiliz ada para obter resultados razoavelmente precisos para os principais parâmetros da camada limite \uf23a\ue7dc \ue1e1 \ue7dc \uebcd \ue1e1 \ue725 \uebd9 \ue1e1 \ue744\uf23b . A ideia aqui é qu e não estamos muito interessados em aprender detalhes da velocidade e d a temperatura na camada limite. Nosso interesse maior é nas suas inclinações na parede, que é onde podemos obter a tensão de cisalhamento , \ue7ec \uebe6 \ued4c \ue003\ue7e4 \uf23a\ue7f2 \ue752 \uebeb \ue200 \ue7f2\ue755 \uf23b \uebec\ueb40\ueb34 , e o fluxo de calor na parede, \ue74d \uebe6 \ued4c \ue003 \ued46\ue747 \uf23a \ue7f2\ue736 \ue200 \ue7f2\ue755 \uf23b \uebec\ueb40\ueb34 , utilizados para o cálculo do coeficiente de atrito e do número de Nusselt. Para usar o m étodo, as equações d a ca mada limite devem ser colocadas na form a integral. Para isso, precisamos integrar as equações na direção y (ond e há os maiores gradientes e onde temos a espessura da camada limite). Os limites de integ r ação s erão: \uf23a\ue372\ue1e1 \ue7dc \uf23b , ou seja, a integração vai da placa até o topo da camada limite . Este método transforma as equações em uma forma ordinária, muito mais simples de serem resolvidas. Apesar das equações na forma integra l não nos revelar nada de novo acerca dos perfis de temperatura e velocidade, ela nos dá equaçõe s com boa precisão acerca de \ue7ec \uebe6 e \ue74d \uebe6 . \u2022 Equação da continuidade \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ued45 \uedb1 \ue7f2 \ue752 \uebec \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ue372 \uedb1 \ue7f2 \ue752 \uebec \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ued46 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 No lado esquerdo temos o teorema do cálculo que diz que a integ ral da derivada de uma função, é igual a própria função. \uedb1 \ue740\ue742 \ue740\ue755 \ue740\ue755 \uebd5 \uebd4 \ued4c \ue742 \uf23a \ue73d \uf23b \ued46 \ue742 \uf23a\ue73e\uf23b Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa Assim: \ued63\ue752 \uebec \ued67 \uebec\ueb40\ueb34 \uebec\ueb40 \uec0b \ued4c \ued46 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ue752 \uebec \ue201 \uebec\ueb40\uec0b \ued46 \ue752 \uebec \ue201 \uebec\ueb40\ueb34 \ued4c \ued46 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 Como o fluido em contato com a placa assume a velocidade da placa (condição de não deslizamento), e ela ou está se movendo paralelamente em relação a corrente livre (na direção \ue754 ) ou está parada, temos que: \ue752 \uebec \ue201 \uebec\ueb40\ueb34 \ued4c \ue372 Então: \u2022 Equação do movimento Considerando \ueacd constante: \uedb1 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \uec0b \ueb34 \ue740\ue755 \ued45 \uedb1 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ueacd \uedb1 \ue7f2\ue37e\ue752 \uebeb \ue7f2\ue755 \ue37e \uec0b \ueb34 \ue740\ue755 Ou: \uedb1 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \uec0b \ueb34 \ue740\ue755 \ued45 \uedb1 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ueacd \uedb1 \ue7f2 \ue7f2\ue755 \uec0b \ueb34 \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \ue740\ue755 Analisando o lado direito, temos novamente o teorema fundamental do cálculo: \uedb1 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \uec0b \ueb34 \ue740\ue755 \ued45 \uedb1 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ueacd \uf248\ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\uec0b \ued46 \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 \uf249 Como a velocidade no topo da camada limite é máxima e sempre ig ual a \ue752 \uebb6 , que é a velocidade da corrente livre (consta nte), pode-se diz er que: \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\uec0b \ued4c \ue372 Assim: \uedb1 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \uec0b \ueb34 \ue740\ue755 \ued45 \uedb1 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ued46\ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 \ue752 \uebec \ue201 \uebec\ueb40\uec0b \ued4c \ued46 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa Agora resolvendo o segundo termo do lado esquerdo por parte s, ao definirmos: \ue097 \ued4c \ue752 \uebec \ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue740\ue098 \ued4c \ue003 \ue7f2 \ue752 \uebeb \ue7f2\ue755 \ue740\ue755 \ue740\ue097 \ued4c \ue7f2 \ue752 \uebec \ue7f2\ue755 \ue740\ue755\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003 \ue098 \ued4c \ue003 \ue752 \uebeb Então: \uedb1 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \uf23e \ue097\ue098 \uf23f \uebec\ueb40\ueb34 \uebec \ueb40\uec0b \ued46 \uedb1 \ue098 \uec0b \ueb34 \ue740\ue097 \uedb1 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ued63 \ue752 \uebeb \ue752 \uebec \ued67 \uebec\ueb40\ueb34 \uebec\ueb40\uec0b \ued46 \uedb1 \ue752 \uebeb \uec0b \ueb34 \ue7f2\ue752 \uebec \ue7f2\ue755 \ue740\ue755 Como já discutido anteriormente, a componente y da velocidade é nula em \ue755 \ued4c \ue372 , fazendo com que o termo relativo ao limite inferior de int egração, no primeir o termo, seja anulado, nos gerando então: \uedb1 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ued6b\ue752 \uebeb \ue201 \uebec\ueb40 \uec0b \ued6f\ued6b\ue752 \uebec \ue201 \uebec\ueb40\uec0b \ued6f \ued46 \uedb1 \ue752 \uebeb \uec0b \ueb34 \ue7f2\ue752 \uebec \ue7f2\ue755 \ue740\ue755 Sabe-se que, a velocidade na direção \ue754 , no topo da camada limite é igual a velocidade da corrente livre. \ue752 \uebeb \ue201 \uebec\ueb40\uec0b \ued4c \ue752 \uebb6 E também relembrando a expressão obtida através da equaçã o da continuidade: \ue752 \uebec \ue201 \uebec\ueb40 \uec0b \ued4c \ued46 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 Temos então que: \uedb1 \ue752 \uebec \ue7f2 \ue752 \uebeb \ue7f2\ue755 \uec0b \ueb34 \ue740\ue755 \ued4c \ued46\ue752 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ued46 \uedb1 \ue752 \uebeb \uec0b \ueb34 \ue7f2\ue752 \uebec \ue7f2\ue755 \ue740\ue755 Substituindo em nossa equação integral do movimento: \uedb1 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \uec0b \ueb34 \ue740\ue755 \ued46 \ue752 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ued46 \uedb1 \ue752 \uebeb \uec0b \ueb34 \ue7f2\ue752 \uebec \ue7f2\ue755 \ue740\ue755 \ued4c \ued46\ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40 \ueb34 Novamente, segundo a equação da continuidade: \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued45 \ue7f2 \ue752 \uebec \ue7f2\ue755 \ued4c \ue372 Logo: Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa \ue7f2 \ue752 \uebec \ue7f2\ue755 \ued4c \ued46 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue003 Substituindo no terceiro termo: \uedb1 \ue752 \uebeb \ue7f2 \ue752 \uebeb \ue7f2\ue754 \uec0b \ueb34 \ue740\ue755 \ued46 \ue752 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ued45 \uedb1 \ue752 \uebeb \uec0b \ueb34 \ue7f2\ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \ued4c \ued46\ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40 \ueb34 Somando o primeiro e segundo termo: \ued46\ue752 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ued45 \uedb1 \ue374\ue752 \uebeb \uec0b \ueb34 \ue7f2\ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \ued4c \ued46\ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Multiplicando tudo por \uf23a\ued46\ue373\uf23b : \ue752 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ued46 \uedb1 \ue374\ue752 \uebeb \uec0b \ueb34 \ue7f2\ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \ued4c \ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Sabendo-se que: \ue740 \uf23a\ue752 \uebeb \ueb36 \uf23b \ue740\ue754 \ued4c \ue752 \uebeb \ue740 \ue752 \uebeb \ue740\ue754 \ued45 \ue752 \uebeb \ue740 \ue752 \uebeb \ue740\ue754 \ued4c \ue374\ue752 \uebeb \ue740 \ue752 \uebeb \ue740\ue754 Temos: \ue752 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ued46 \uedb1 \ue7f2 \uf23a\ue752 \uebeb \ueb36 \uf23b \ue7f2\ue754 \uec0b \ueb34 \ue740\ue755 \ued4c \ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Juntando as integrais do lado esquerdo: \uedb1 \uf248\ue752 \uebb6 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued46 \ue7f2 \uf23a\ue752 \uebeb \ueb36 \uf23b \ue7f2\ue754 \uf249 \ue740\ue755 \uec0b \ueb34 \ued4c \ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Ou, como \ue752 \uebb6 é constante: \uedb1 \uf248 \ue7f2\ue752 \uebb6 \ue752 \uebeb \ue7f2\ue754 \ued46 \ue7f2 \uf23a\ue752 \uebeb \ueb36 \uf23b \ue7f2\ue754 \uf249 \ue740\ue755 \uec0b \ueb34 \ued4c \ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 A soma das derivadas é igual a derivada da soma: \uedb1 \ue7f2 \ue7f2\ue754 \uf23e \ue752 \uebb6 \ue752 \uebeb \ued46 \ue752 \uebeb \ueb36 \uf23f \ue740\ue755 \uec0b \ueb34 \ued4c \ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 \uedb1 \ue7f2 \ue7f2\ue754 \uf23e \ue752 \uebb6 \ue1e4 \ue752 \uebeb \ued46 \ue752 \uebeb \ue1e4 \ue752 \uebeb \uf23f \ue740\ue755 \uec0b \ueb34 \ued4c \ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 \uedb1 \ue7f2 \uf23e \uf23a\ue752 \uebb6 \ued46 \ue752 \uebeb \uf23b\ue752 \uebeb \uf23f \ue7f2\ue754 \ue740\ue755 \uec0b \ueb34 \ued4c \ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Podemos rearranjar essa equação observando que: Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa \uedb1 \ue7f2\ue742 \uf23a \ue754\ue1e1 \ue755 \uf23b \ue7f2\ue754 \ue740\ue755 \uebd5 \uebd4 \ued4c \ue740 \ue740\ue754 \uedb1 \ue742 \uf23a \ue754\ue1e1 \ue755 \uf23b \ue740\ue755 \uebd5 \uebd4 Ex emplo: \ue742 \uf23a \ue754 \ue1e1 \ue755 \uf23b \ued4c \ue374\ue754\ue37e\ue755\ue37e Prova: \uedb1 \ue376 \ue754\ue755 \ue37e \uebd5 \uebd4 \ued4c \ue740 \ue740\ue754 \uedb1 \ue374\ue754\ue37e\ue755\ue37e\ue740\ue755 \uebd5 \uebd4 \uedb1 \ue376 \ue754\ue755 \ue37e \uebd5 \uebd4 \ued4c \ue740 \ue740\ue754 \uf248\ue374\ue754\ue37e \uedb1 \ue755\ue37e\ue740\ue755 \uebd5 \uebd4 \uf249 \uedb1 \ue376 \ue754\ue755 \ue37e \uebd5 \uebd4 \ued4c \ue376\ue754 \uedb1 \ue755\ue37e\ue740\ue755 \uebd5 \uebd4 \ued45 \ue374\ue754\ue37e \ue740 \ue740\ue754 \uedb1 \ue755\ue37e\ue740\ue755 \uebd5 \uebd4 \uedb1 \ue376 \ue754\ue755 \ue37e \uebd5 \uebd4 \ued4c \uedb1 \ue376 \ue754\ue755 \ue37e\ue740\ue755 \uebd5 \uebd4 Então: A deriva da ordinária, e n ão mais parcial, se explica, pois, uma integra l definida em y, de fun ção do tipo \ue742\uf23a \ue754\ue1e1 \ue755 \uf23b , gera uma função \ue728 \uf23a\ue754 \uf23b . Essa equação é a forma int egral da equação do momento da camada limite. Analogamente para a equação da energia : Aplicando a equação da continuidade para a camada limite de temperatura: Integrando a equação da energia: \uedb1 \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued45 \uedb1 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ue8bb \uedb1 \ue7f2 \ue7f2\ue755 \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uec0b \uecc5 \ueb34 \ue740\ue755 \ue740 \ue740\ue754 \uf248\uedb1 \uf23a\ue752 \uebb6 \ued46 \ue752 \uebeb \uf23b\ue752 \uebeb \ue740\ue755 \uec0b \ueb34 \uf249 \ued4c \ueacd \ued6c \ue7f2\ue752 \uebeb \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 \ue752 \uebec \ue201 \uebec\ueb40\uec0b \uecc5 \ued4c \ued46 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa \uedb1 \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued45 \uedb1 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ue8bb \uf248\ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\uec0b \uecc5 \ued46 \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 \uf249 Semelhantemente à equação do movimento, temos que \uf240 \uec21\uebcd \uec21\uebec \uf241 \uebec \ueb40\uec0b \uecc5 \ued4c \ue372 pois a temperatura nesse ponto é constante e igual a temperatura da corre nte livre. \uedb1 \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued45 \uedb1 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ued46\ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Integrando o segundo termo por partes: \ue097 \ued4c \ue752 \uebec \ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue740\ue098 \ued4c \ue003 \ue7f2\ue736 \ue7f2\ue755 \ue740\ue755 \ue740\ue097 \ued4c \ue7f2 \ue752 \uebec \ue7f2\ue755 \ue740\ue755\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003\ue003 \ue003\ue098 \ued4c \ue003\ue736 Então: \uedb1 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \uf23e \ue097\ue098 \uf23f \uebec\ueb40 \ueb34 \uebec\ueb40\uec0b \uecc5 \ued46 \uedb1 \ue098 \uec0b \uecc5 \ueb34 \ue740\ue097 \uedb1 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ued63\ue752 \uebec \ue736 \ued67 \uebec\ueb40\ueb34 \uebec\ueb40\uec0b \uecc5 \ued46 \uedb1 \ue736 \uec0b \uecc5 \ueb34 \ue7f2 \ue752 \uebec \ue7f2\ue755 \ue740\ue755 Como \ue752 \uebec \ue201 \uebec\ueb40\ueb34 \ued4c \ue372 , o limite inferior do primeiro termo da direita se anula, nos gerando: \uedb1 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ue752 \uebec \ue201 \uebec\ueb40\uec0b \uecc5 \ue736 \ue201 \uebec \ueb40\uec0b \uecc5 \ued46 \uedb1 \ue736 \uec0b \uecc5 \ueb34 \ue7f2 \ue752 \uebec \ue7f2\ue755 \ue740\ue755 Sabe-se também que no topo da camada limite: \ue736\ue201 \uebec \ueb40\uec0b \uecc5 \ued4c \ue736 \uebb6 Além disso, a equação ob tida na e quação da continuidade, diz: \ue752 \uebec \ue201 \uebec\ueb40 \uec0b \uecc5 \ued4c \ued46 \uedb1 \ue7f2\ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 Aplicando essas duas condições: \uedb1 \ue752 \uebec \ue7f2\ue736 \ue7f2\ue755 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ued46\ue736 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued46 \uedb1 \ue736 \uec0b \uecc5 \ueb34 \ue7f2 \ue752 \uebec \ue7f2\ue755 \ue740\ue755 Substituindo esta expressão em nossa equaçã o integral da energia: \uedb1 \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued46 \ue736 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued46 \uedb1 \ue736 \uec0b \uecc5 \ueb34 \ue7f2 \ue752 \uebec \ue7f2\ue755 \ue740\ue755 \ued4c \ued46\ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Transferência de Calor \u2013 José Sabino e Pr iscilla Arouxa Utilizando a equação da continuidade, sabe-se que: \ue7f2 \ue752 \uebec \ue7f2\ue755 \ued4c \ued46 \ue7f2 \ue752 \uebeb \ue7f2\ue754 Então: \uedb1 \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued46 \ue736 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued45 \uedb1 \ue736 \uec0b \uecc5 \ueb34 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \ued4c \ued46\ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Agrupando a primeira e a terceira integral: \ued46\ue736 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued45 \uedb1 \ued64\ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ued45 \ue736 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued68 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ued46\ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Sendo \ue752 \uebeb \ue7f2\ue736 \ue7f2\ue754 \ued45 \ue736 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued4c \ue7f2 \ue736\ue752 \uebeb \ue7f2\ue754 \ue003 Temos: \ued46\ue736 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued45 \uedb1 \ue7f2 \ue736\ue752 \uebeb \ue7f2\ue754 \ue003\ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ued46\ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Ou: \ue736 \uebb6 \uedb1 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued46 \uedb1 \ue7f2 \ue736\ue752 \uebeb \ue7f2\ue754 \ue003\ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Agrupando, \uedb1 \ued64\ue736 \uebb6 \ue7f2 \ue752 \uebeb \ue7f2\ue754 \ued46 \ue7f2 \ue736\ue752 \uebeb \ue7f2\ue754 \ued68 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Como \ue736 \uebb6 é constante: \uedb1 \ued64 \ue7f2 \ue736 \uebb6 \ue752 \uebeb \ue7f2\ue754 \ued46 \ue7f2 \ue736\ue752 \uebeb \ue7f2\ue754 \ued68 \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Aplicando a regra da soma das derivadas: \uedb1 \ue7f2 \ue7f2\ue754 \uf23e \ue736 \uebb6 \ue752 \uebeb \ued46 \ue736\ue752 \uebeb \uf23f \ue740\ue755 \uec0b \uecc5 \ueb34 \ued4c \ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34 Por fim, explicitando \ue752 \uebeb : \ue740 \ue740\ue754 \uf248\uedb1 \uf23a \ue736 \uebb6 \ued46 \ue736 \uf23b \ue752 \uebeb \ue740\ue755 \uec0b \uecc5 \ueb34 \uf249 \ued4c \ue8bb \ued6c \ue7f2\ue736 \ue7f2\ue755 \ued70 \uebec\ueb40\ueb34

Método Integral Placa plana Escoamento paralelo

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Fenômenos de Transporte I

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