Lecture 4 1/28/2019. CM3120 Transport/Unit Operations 2

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1 CM3120 ransport/unit Operations 2 State Heat ransfer Professor Faith Morrison Department of Chemical Engineering Michigan echnological University wwwchemmtuedu/~fmorriso/cm3120/cm3120html 1 o get started, let s contrast the steady and unsteady cases in a familiar problem: Steady HE RNSFER vs 1D, heat conduction, in the direction, in a slab 2 1

2 1D, rectangular geometry: Independent of time Flu constant linear temperature profile Steady resistance to heat transfer at both boundaries Heat ransfer at Steady State (Newton s law of cooling Cs) Resistance to heat transfer h q h H h What s the answer? Resistance to heat transfer h W 3 1D, rectangular geometry: Independent of time Flu constant linear temperature profile Steady resistance to heat transfer at both boundaries Heat ransfer at Steady State (Newton s law of cooling Cs) Resistance to heat transfer h q h H h Resistance to heat transfer h W 4 2

3 emperature distribution: Resistance to heat transfer h Heat ransfer at Steady State (Newton s law of cooling Cs) Resistance to heat transfer h q h k d d h Flu constant Independent of time 5 Heat ransfer here are many circumstances that cause unsteady heat transfer o imagine a case where heat transfer is unsteady: We must specify the state of the system at some point in time (initial conditions) We must specify what then happens to cause heat to start to transfer (the scenario) 6 3

4 Can you think of any real situations? Can you write them in terms of: initial conditions and a modeling scenario? You try 7 0 Heat ransfer Eample: wide, tall slab initially at is suddenly subjected to flowing fluid on its two broad faces he left fluid is at and its heat transfer to the wall is characterized by heat transfer coefficient h, while the right side is at and characterized by h What is the temperature distribution across the slab as a function of time? What do we think will happen? Will there be heat transfer resistance at the boundaries? Will there be a linear temperature profile in the slab? Femtoseconds after the change, what does the profile look like? What will the solution trend towards as time goes on? 8 4

5 Heat ransfer t10 s Eample: wide, tall slab initially at is suddenly subjected to flowing fluid on its two broad faces he left fluid is at and its heat transfer to the wall is characterized by heat transfer coefficient h, while the right side is at and characterized by h What is the temperature distribution across the slab as a function of time? What do we think will happen? You try 0 9 Heat ransfer Eample: wide, tall slab initially at is suddenly subjected to flowing fluid on its two broad faces he left fluid is at and its heat transfer to the wall is characterized by heat transfer coefficient h, while the right side is at and characterized by h What is the temperature distribution across the slab as a function of time? h 0 h t10 s What do we think will happen? Will there be heat transfer resistance at the boundaries? Will there be a linear temperature profile in the slab? Femtoseconds after the change, what does the profile look like? What will the solution trend towards as time goes on? 10 5

6 t10 s q q q h q q q q 0 h Heat flows in the positive direction 11 t10 s q q q h What do we think will happen? What s net? q q Heat flows in the positive direction q q 0 h 12 6

7 q tt q q and varying with t q h q, q h, q f,t 13 q tt q q and varying with t q h q, q h, q f,t 14 7

8 q tt q q and varying with t q h q, q h, q f,t 15 Steady q t q q q and constant q h q, q h, constant 16 8

9 Steady decreases also decreases 17 Steady decreases also decreases Q: Does it always happen like this? 18 9

10 Steady decreases also decreases Q: Does it always happen like this? : Mmmm No It depends 19 What are the various cases that are seen? h are large Let s try

11 What are the various cases that are seen? k is large Let s try 0 21 What are the various cases that are seen? Neither slab conduction nor fluid convection dominates Let s try

12 What are the various cases that are seen? If h is large, the wall temp is just the bulk temperature (fast convection) If k is large, the temp profile is always straight (quasi steady state in the slab) and the convection works to keep up (heat transfer is limited by h ; fast conduction in slab) If neither mechanism dominates, it s complicated! If the boundary conditions vary with t,, it s complicated! decreases also decreases 23 What is our usual strategy for comple phenomena? nswer: Dimensional nalysis CM3110: Momentum and Heat Xfer Comple Heat ransfer Dimensional nalysis Eperience with Dimensional nalysis (momentum): Flow in pipes at all flow rates (laminar and turbulent) Solution: Navier-Stokes, Re, Fr, L/D, dimensionless wall force f; ffre,l/d Rough pipes Solution: add additional length scale; then nondimensionalize Let s review Non-circular conduits Solution: Use hydraulic diameter as the length scale of the flow to nondimensionalize Flow around obstacles (spheres, other comple shapes Solution: Navier-Stokes, Re, dimensionless drag C ; C C Re oundary layers Solution: wo components of velocity need independent lengthscales 24 12

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