CM3120 Module 2 Lecture I 1/4/2021fmorriso/cm3120/lectures/2021_CM3120_modu… · Unsteady State Heat Transfer I. Introduction II. Unsteady Microscopic Energy Balance—(slash and

CM3120 Module 2 Lecture I 1/4/2021

1

CM3120: Module 2

© Faith A. Morrison, Michigan Tech U.1

Unsteady State Heat Transfer

I. IntroductionII. Unsteady Microscopic Energy Balance—(slash and burn)III. Unsteady Macroscopic Energy BalanceIV. Dimensional Analysis (unsteady)—Biot number, Fourier

numberV. Low Biot number solutions—Lumped parameter analysisVI. Short Cut Solutions—(initial temperature 𝑇 ; finite ℎ),

Gurney and Lurie charts (as a function of position, 𝑚Bi, and Fo); Heissler charts (center point only, as a function of 𝑚 1/Bi, and Fo)

VII. Full Analytical Solutions (stretch)

CM3120: Module 2


Professor Faith A. Morrison

Department of Chemical EngineeringMichigan Technological University

www.chem.mtu.edu/~fmorriso/cm3120/cm3120.html

Lecture I:Introduction to Unsteady State

Heat Transfer


2


HEAT TRANSFERSteady

Heat Transfer: Steady vs. Unsteady

Unsteadyvs.

To get started, let’s contrast the steady and unsteady cases in a familiar problem:

1D, heat conduction, in the 𝒙‐direction,

in a slab

Newton’s law of Cooling boundary conditions

𝑇

𝑇

𝑇 𝑇

1D, rectangular geometry:• Independent of time• Flux constant• linear temperature profile• Steady resistance to heat

transfer at both boundaries


Heat Transfer at Steady State

Aqx H

W

B x𝑥

𝑇

B

𝑇

𝑇

𝑇 𝑇

𝑇ℎ

𝑇ℎ

(Newton’s law of cooling BCs)

.

.𝑇

𝑇

Resistance to heat transfer

1/ℎ

Resistance to heat transfer 1/ℎ

What’s the temperature

profile?



3


Heat Transfer at Steady State

Aqx H

W

B x𝑥

𝑇

B

𝑇

𝑇

𝑇 𝑇

𝑇ℎ

𝑇ℎ

(Newton’s law of cooling BCs)

.

.𝑇

𝑇


1/ℎ


1D, rectangular geometry:• Independent of time• Flux constant• linear temperature profile• Steady resistance to heat

transfer at both boundaries



Heat Transfer at Steady State(Newton’s law of cooling BCs)

𝑥

𝑇

B

𝑇

𝑇 .

.𝑇

𝑇


1/ℎ


𝑞𝐴 ℎ 𝑇 𝑇 ℎ 𝑇 𝑇𝑘

𝑑𝑇𝑑𝑥

Flux constant

Temperature distribution:

Independent of time

𝑇 𝑥

𝑇 𝑇 1ℎ𝑞𝐴

Newton’s law BC:



4


Unsteady Heat Transfer

• 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).

There are many circumstances that cause unsteady heat transfer.

To imagine a case where heat transfer is unsteady:



Can you think of any real engineering situations (unsteady heat xfer)?Can you write them in terms of:

• initial conditions and • a modeling scenario? You try.


5



𝑥

𝑇

B

.

.𝑇

𝑇

Example: A wide, tall slab initially uniformly at 𝑇is suddenly subjected to flowing fluid on its two broad faces. The left fluid is at 𝑇 and its heat transfer to the wall is characterized by heat transfer coefficient ℎ , while the right side is at 𝑇and 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 → ∞ ?

𝑇

0




𝑥

𝑇

B

.

.𝑇

𝑇

What do we think will happen?

You try.𝑇

0

𝑡 10 𝑠

𝑓𝑒𝑚𝑡𝑜𝑠𝑒𝑐𝑜𝑛𝑑 10 𝑠

Example: A wide, tall slab initially uniformly at 𝑇is suddenly subjected to flowing fluid on its two broad faces. The left fluid is at 𝑇 and its heat transfer to the wall is characterized by heat transfer coefficient ℎ , while the right side is at 𝑇and characterized by ℎ . What is the temperature distribution across the slab as a function of time?



6



𝑇

.

.𝑇

𝑇

𝑇𝑡 10 𝑠

𝑥B

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 → ∞ ?0

• Femtoseconds after the change

ℎ

ℎ

Unsteady Heat TransferExample: A wide, tall slab initially uniformly at 𝑇is suddenly subjected to flowing fluid on its two broad faces. The left fluid is at 𝑇 and its heat transfer to the wall is characterized by heat transfer coefficient ℎ , while the right side is at 𝑇and characterized by ℎ . What is the temperature distribution across the slab as a function of time?



𝑞𝐴ℎ 𝑇 𝑇



B 𝑥

𝑇

.

.𝑇

𝑇

𝑇

𝑡 10 𝑠

𝑞𝐴

𝑞𝐴

𝑞𝐴 0

Unsteady

Heat flows in the positive 𝑥‐ direction →

𝑇 𝑇 1ℎ𝑞𝐴

Newton’s law BC:

𝑞𝐴 ∝

∝ 𝑞𝐴


7




B 𝑥

𝑇

.

.𝑇

𝑇

𝑇

𝑡 10 𝑠

𝑞𝐴

𝑞𝐴

𝑞𝐴 0

Unsteady

Heat flows in the positive 𝑥‐ direction →

What do we think will happen?

What’s next?

𝑞𝐴 ∝

∝ 𝑞𝐴


𝑞𝐴

𝑞𝐴


B 𝑥

𝑇

.

.𝑇

𝑇

𝑇

𝑡 𝑡

𝑞𝐴ℎ 𝑇 𝑇 , 𝑞𝐴

ℎ 𝑇 , 𝑇

𝑞𝐴 𝑓 𝑥, 𝑡

and varying with 𝑡

Unsteady

𝑞𝐴 ∝

∝ 𝑞𝐴



8


B 𝑥

𝑇

.

.𝑇

𝑇

𝑇

𝑡 𝑡


ℎ 𝑇 , 𝑇


Unsteady

𝑞𝐴 ∝

∝ 𝑞𝐴

𝑞𝐴

𝑞𝐴




B 𝑥

𝑇

.

.𝑇

𝑇

𝑇

𝑡 𝑡


ℎ 𝑇 , 𝑇


Unsteady

𝑞𝐴 ∝

∝ 𝑞𝐴

…

𝑞𝐴

𝑞𝐴




9


B 𝑥

𝑇

.

.𝑇

𝑇

𝑇

𝑡 ∞


ℎ 𝑇 , 𝑇

constant

𝑞𝐴

𝑞𝐴

𝑞𝐴𝑇

𝑇

Steady

𝑞𝐴 ∝

∝ 𝑞𝐴and constant


© F

aith

A. M

orris

on, M

ichi

gan

Tech

U.

18B 𝑥

𝑇

.

.𝑇

𝑇

𝑇decreases with time

also decreases with time

SteadyUnsteady


10

© F

aith

A. M

orris

on, M

ichi

gan

Tech

U.

19B 𝑥

𝑇

.

.𝑇

𝑇



SteadyUnsteady

Q: Does it always happen like this?

© F

aith

A. M

orris

on, M

ichi

gan

Tech

U.

20B 𝑥

𝑇

.

.𝑇

𝑇



SteadyUnsteady

Q: Does it always happen like this?

A: Mmmm. No. It depends. . . .


11


What are the various cases that are seen?

𝑥

𝑇

B

.

.𝑇

𝑇

𝑇

0

Let’s think.

• ℎ are large



• ℎ are largeHeat Transfer: Steady vs. Unsteady

𝑥

𝑇

B

.

.𝑇

𝑇

𝑇

0


12


What are the various cases that are seen?• 𝑘 is large

𝑥

𝑇

B

.

.𝑇

𝑇

𝑇

0

Let’s think.



• 𝑘 is largeHeat Transfer: Steady vs. Unsteady

𝑥

𝑇

B

.

.𝑇

𝑇

𝑇

0


13


What are the various cases that are seen?• Neither slab conduction

nor fluid convection dominates

𝑥

𝑇

B

.

.𝑇

𝑇

𝑇

0

Let’s think.



• Neither slab conduction nor fluid convection dominates


𝑥

𝑇

B

.

.𝑇

𝑇

𝑇

0


14

B 𝑥

𝑇

.

.𝑇

𝑇




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, 𝑥, it’s complicated!



Documents

CM3120 Module 2 Lecture I 1/4/2021fmorriso/cm3120/lectures/2021_CM3120_modu… · Unsteady State Heat Transfer I. Introduction II. Unsteady Microscopic Energy Balance—(slash and