Question

PLEASE DO NOT REPOST THE ANSWER THAT ANOTHER TUTOR POSTED. THAT ANSWER IS WRONG. PLEASE I NEED HELP, THANK YOU

Transient Conduction 1. (10 pts) The corner of a rectangular object experiences convection on its left and top sides, as shown below. Initially, the object experiences no heat generation and has a steady uniform temperature of Ti. Suddenly, at t 0, a chemical reaction takes place throughout the volume of the material, which causes heat generation (q"). Using the energy balance method, derive the finite-difference equation for the temperature of node 1,T,and node 5,Ts, as a function of time. Use the implicit method to describe the time dependence. You can assume the medium has a thermal conductivity of k. You must draw the CV and relevant heat vectors for BOTH nodes to receive full credit. 2 5 Те, Ta Ay

Answer 1

SOLUTION:

We are given a 2-D transient heat conduction problem with heat generation. Here, the 2D plate is exposed to convection heat transfer on its two edges and we have to find the temperature at two nodes one ( Node 1) at the exterior corner of the plate with convection and another ( Node 5) is the interior node.

2 To AX

Assumptions:

• The conduction heat transfer coefficient is k.

Analysis:

Fest the node 1 マ-D conduction toe tave eqyaion with.fectc,enentd.,or?' eql1) To dig tit#2e the pmoblem integer puch that in time, wen knocldce he-7ean VC. ㄒ OT 2군 m,n egn(2) An nude 1, n addhhon to conducton heat tnany ese, thene which Mimity the upe ea () hene we ll ve the eneJijy balance mehod hene, 2. 2

萷milanu,, 1 2. △t Ic K2 Solving foor TiP1, we have, aj丁/cmz , we can uste the above e9nag:- k.

Node S. tene. But we method aus qivert below 2. △沈 1 Now, we fave dlenived e dee equat node 1 and nocle 5. Note #1: These e9uaHon ane numea&cully cnAtable and 42

3tabili.ty csửřenon : Nocle 1: Wode 5 7+ These two values anbe evaltedut by olperuneg eg4) Note*2: 꺼efeat gone.nation, q" my be a me

Limitations:

• In the explicit finite difference method used here, we can find the temperature of any node at a time $t+\Delta t$ with the knowledge of the temperature of its surrounding nodes for the preceding time t.  
• Hence, in this method determination of the temperature of any node is independent of the temperature of other nodes at the same time.
• Although this method is numerically convenient, it suffers from the limitation of selection of $\Delta t$ .
• Since for the stability reasons, we have to use an extremely small value of $\Delta t$ , it results in a very large number of time intervals for evaluation.