This chapter provides a comprehensive exploration of steady-state conduction, a fundamental mode of heat transfer in which temperature distributions within a solid remain constant over time. The document delves into the underlying principles, mathematical models, and practical applications of stead...
Forsberg Heat Transfer
Chapter 3
Steady-State Conduction
One-Dimensional Conduction
Heat Flow Through a Plane Wall
T T T T
k + k + k + qgen = c
x x y y z z t
1-D, steady, no heat generation, constant k
d 2T
Equation reduces to: =0
dx 2
1
, Heat Flow Through a Plane Wall (Cont’d)
BC: At x = 0, T = T1
T1 = C1 (0) + C 2 C 2 = T1
At x = L. T = T2
2
dT T2 = C1L + C 2 C1 = (T2 − T1 ) / L
=0
dx 2
T −T
dT T (x) = 2 1 x + T1
= C1 L
dx
dT kA
T (x) = C1 x + C 2 qx = −k Ax = (T1 − T2 )
dx L
Multilayered Walls
k1 A k2 A k3 A k4 A
q= (T1 − T2 ) q= (T2 − T3 ) q= (T3 − T4 ) q= (T4 − T5 )
L1 L2 L3 L4
L L L L
q 1 = (T1 − T2 ) q 2 = (T2 − T3 ) q 3 = (T3 − T4 ) q 4 = (T4 − T5 )
k1 A k2 A k3 A k4 A
Summing the left and right sides of these four equations, and rearranging
T1 − T5
q =
L1 L L L
+ 2 + 3 + 4
k1 A k2 A k3 A k4 A
2
, Electric-Heat Analogy and Resistance Concept
E
Ohm's Law: I =
R
(I = current, E = potential causing current flow, R = resistance)
kA
Heat flow through a plane wall: q = (T )
L
T
Rearranging, we get q =
L
kA
Comparing this to Ohm's Law
L
q I T E R
kA
L
So, is the thermal resistance to heat flow
kA
for a plane wall
Back to Multilayered Walls
T1 − T5
q =
L1 L L L
+ 2 + 3 + 4
k1 A k2 A k3 A k4 A
T1 − T5 is the overall temperature difference across the wall. This
is causing the heat flow.
The denominator is the sum of the resistances between the two
temperatures.
Toverall
This can be generalized to q=
R
3
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller AhmedKnight. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.49. You're not tied to anything after your purchase.
