Biosystems Engineering Lecture 3: Thermal Energy Transport in 1-D Systems - Prof. Glenn Br, Study notes of Heat and Mass Transfer

Download Biosystems Engineering Lecture 3: Thermal Energy Transport in 1-D Systems - Prof. Glenn Br and more Study notes Heat and Mass Transfer in PDF only on Docsity! Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 1 BAE 3013 Lecture 3 Today • Differential control volume analysis of heat transport. Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 2 Control Volume (1-D) Word equation: rate heat in -rate heat out +heat source = change in thermal energy stored Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 3 Heat flux in: "" xx eq + Conduction + Convection Total heat in: tzyeq xxx ∆∆∆+ )""( Total heat out: tzyeq xxxx ∆∆∆+ ∆+)""( Heat Sources tzyxQ ∆∆∆∆ Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 4 Q = volumetric heat sources (W/m3) Sources include biological & chemical reactions, radiation, and external heat not considered in q” and e” Heat energy in storage zyxTcvolumeU p ∆∆∆=∗ ρ For change in heat energy storage, replace T with ∆T zyxTc p ∆∆∆∆ρ Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 5 Put it all together tzyeqtzyeq xxxxxxx ∆∆∆+−∆∆∆+ ∆+)""()""( TzyxctzyxQ p ∆∆∆∆=∆∆∆∆+ ρ Divide by ∆x∆y∆z∆t and rearrange t TcQ x ee x qq p xxxxxxxxxx ∆ ∆ =+ ∆ − − ∆ − − ∆+∆+ ρ """" Notice how the sign has been selected. Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 6 Let 0& →∆∆ tx t TcQ x e x q p xx ∂ ∂ =+ ∂ ∂ − ∂ ∂ − ρ"" Let qx” be defined by Fourier’s Law. x Tkqx ∂ ∂ −=" Let ex” be defined by: Tcue px ρ=" Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 7 ( ) t TcQuTc xx Tk x pp ∂ ∂=+ ∂ ∂−      ∂ ∂ ∂ ∂ ρρ With k, cp and ρ constant ( ) t T c Q x uT x T c k pp ∂ ∂ =+ ∂ ∂ − ∂ ∂ ρρ 2 2 Conduction Convection Source Storage Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 8 1-D Steady State ( ) 0→− ∂ ∂ t ( ) 02 2 =+− pp c Q dx uTd dx Td c k ρρ 1-D; Steady state, No Sources, Q = 0 ( ) 02 2 =− uT dx dc dx Tkd pρ Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 9 In 3-D, transient, with sources t T c Q z wT y T x uT z T y T x T c k p p ∂ ∂ =+       ∂ ∂ + ∂ ∂ + ∂ ∂ −      ∂ ∂ + ∂ ∂ + ∂ ∂ ρ ν ρ 2 2 2 2 2 2 Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 10 Question: What type of equation is the following? ( ) t T c Q x uT x T c k pp ∂ ∂ =+ ∂ ∂ − ∂ ∂ ρρ 2 2 Ordinary or partial? Order? Linear or non-linear? Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 11 Question: What order are each of the terms on the left hand side (LHS)? ( ) pp c Q x uT x T c k ρρ + ∂ ∂ − ∂ ∂ 2 2 Oklahoma State University Biosystems Engineering BAE 3013 Lecture 3 12 Question: How many initial and/or boundary conditions do we need to solve this equation? ( ) t T c Q x uT x T c k pp ∂ ∂ =+ ∂ ∂ − ∂ ∂ ρρ 2 2