This document presents an experiment on heat transfer linear conductivity and effect conducted by students at the Polytechnic University of Puerto Rico. The objectives were to determine the thermal conductivity and temperature difference in various materials, study contact resistance, and analyze resistance effects. The theory section covered heat transfer concepts like Fourier's law, thermal conductivity, and thermal contact resistance. The procedure measured temperature distributions and heat flow through uniform and composite walls to calculate conductivity. Safety and references were also provided.

1. Polytechnic University of Puerto Rico
Chemical Engineering Department
Winter 2016
Ashley M. Ramirez Quiles
Stephanie P. Rivera Ares
Edgared M. Troche Muñiz
Jorge Sepulveda
November 29, 2016
CHE 3321- 22
Prof. Pablo Traverso
Heat Transfer Linear Conductivity
Effect

2. Agenda
Objectives
Theory
Equipment
Procedure
Data
Calculations
Security
References

3. Objectives
 Determine thermal conductivity and temperature
difference in stainless steel, brass, aluminum and
isolators for a steady state conduction.
 Contact resistance effect in the interphase and thermal
paste
 Conduct analysis and comparison of the resistance
effects

4. Theory
Conduction Heat Transfer
 Energy is transferred as heat due to a temperature potential difference in a body, or
between bodies in contact with each other.
Figure 1. Representation of heat flow [1]

5. Theory
Thermal proportionality constant (𝒌):
𝑞 𝑥
𝐴
~
𝜕𝑇
𝜕𝑥
; Heat transfer rate is proportional to the temperature gradient.
 𝑞 𝑥 = −𝑘𝐴
𝜕𝑇
𝜕𝑥
; Fourier’s Law
 Wiedemann-Franz Law:
𝐿 =
𝑘
𝜎𝑇
, and 𝑘 = σLT
Figure 2. One-dimensional
steady state heat transfer,
temperature gradients. [2]

6. Theory
Energy Balance:
𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 𝐸𝑖𝑛 − 𝐸 𝑜𝑢𝑡 + 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
 𝑞 𝑥 + 𝑞 𝑦 + 𝑞 𝑧+𝑞 𝑔𝑒𝑛= 𝑞 𝑥+𝑑𝑥+𝑞 𝑦+𝑑𝑦 + 𝑞 𝑧+𝑑𝑧 +
𝑑𝐸
𝑑τ
For constant Thermal Conductivity:

𝜕2 𝑇
𝜕𝑥2 +
𝜕2 𝑇
𝜕𝑦2+
𝜕2 𝑇
𝜕𝑧2 +
𝑞
𝑘
=
1
𝛼
𝜕𝑇
𝜕𝑡
; 𝛼 =
𝑘
𝜌𝐶 𝑝
(
𝑚2
𝑠
)
Figure 2. Volume element for one
dimensional heat-conduction. [2]
T1>T2

7. Theory
 Under steady state conditions, the temperature distribution is linear, and the temperature
gradient may be expressed as:
𝜕2 𝑇
𝜕𝑥2 = 0
𝑑𝑇
𝑑𝑥
= 𝑚
𝑇 = 𝑚𝑥 + 𝑏
Fourier’s Rate Equation;
Experimental 𝑠𝑙𝑜𝑝𝑒 𝑚 =
Δ𝑇
Δ𝑥
; 𝑘 𝑒𝑥𝑝 =
𝑞′′
𝑠𝑙𝑜𝑝𝑒
𝑞 𝑥
𝐴
= 𝑘
(𝑇1−𝑇2)
∆𝑥
;
𝑞 𝑥 = −𝑘𝐴
𝑑𝑇
𝑑𝑥
0
∆𝑥
−𝑞 𝑥
𝑘𝐴
𝑑𝑥 =
𝑇1
𝑇2
𝑑𝑇
Figure 3. Linear direction
of heat flow. [3]

8. Theory
Heat Flow:
𝐻𝑒𝑎𝑡 𝐹𝑙𝑜𝑤 =
𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
= −
∆𝑇𝑜𝑣𝑒𝑟𝑎𝑙𝑙
𝑅𝑡ℎ
Figure 4. One-dimensional heat transfer through a composite
wall, and its equivalent electrical analog. [3]
𝑅𝑡ℎ =
∆𝑥 𝐴
𝑘 𝐴 𝐴
+
∆𝑥 𝐵
𝑘 𝐵 𝐴
+
∆𝑥 𝐶
𝑘 𝐶 𝐴
𝑎𝑛𝑑;
( 𝐴 ∗ 𝑅𝑡ℎ) = 𝑅 𝑣𝑎𝑙𝑢𝑒 =
∆𝑇
𝑞
𝐴
=
∆𝑥
𝑘

9. Theory
Overall Heat-Transfer Coefficient (U)
𝑈 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 = 𝑅 𝑣𝑎𝑙𝑢𝑒
−1
In a composite wall we can state that;
𝑞 =
𝑇𝐴 − 𝑇𝐵
𝑅1 + 𝑅2 + 𝑅3
=
∆𝑇𝑜𝑣𝑒𝑟𝑎𝑙𝑙
𝐴 ∗ 𝑅𝑡ℎ
=
∆𝑇𝑜𝑣𝑒𝑟𝑎𝑙𝑙
𝑅 𝑣𝑎𝑙𝑢𝑒
= UA∆𝑇𝑜𝑣𝑒𝑟𝑎𝑙𝑙
Figure 5. Composite wall. [4]

10. Theory
Reduced cross-sectional area of heat transfer between thermocouples:
From Fourier’s Law;
𝑞 𝑥 = −𝑘𝐴
Δ𝑇
Δ𝑥
Rearranging the equation;
∆𝑇 =
𝑞 𝑥∆𝑥
𝑘𝐴
; ∆𝑇 ∝
1
𝐴
Figure 6. Composite wall with reduced area. [5]

11. Theory
Thermal Contact Resistance:
From the energy balance equation;
𝑞 =
𝑇1−𝑇2𝐴
∆𝑥 𝐴
𝑘 𝐴 𝐴
=
𝑇2𝐴−𝑇2𝐵
1
ℎ 𝑐 𝐴
=
𝑇2𝐵−𝑇3
∆𝑥 𝐵
𝑘 𝐵 𝐴
ℎ 𝑐 = 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
1
ℎ 𝑐 𝐴
= 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
Figure 8. Joint-roughness model for the
analysis of hc . [3]
𝑞 =
𝑇2𝐴 − 𝑇2𝐵
𝐿 𝑔
2𝑘 𝐴 𝐴 𝑐
+
𝐿 𝑔
2𝑘 𝐵 𝐴 𝑐
+ 𝑘 𝑓 𝐴 𝑣
𝑇2𝐴 − 𝑇2𝐵
𝐿 𝑔
=
𝑇2𝐴 − 𝑇2𝐵
1
ℎ 𝑐 𝐴
 ℎ 𝑐 =
1
𝐿 𝑔
𝐴 𝑐
𝐴
2𝑘 𝐴 𝑘 𝐵
𝑘 𝐴+𝑘 𝐵
+
𝐴 𝑣
𝐴
𝑘 𝑓 ;
Figure 7. Illustration of the thermal contact
resistance effect, temperature profile. [5]
Figure 9. Ideal and actual
thermal contact. [2]

12. Theory
Figure 11. Effect of metallic coatings on
thermal contact surfaces. [2]
Figure 10. Contact conductance of typical surfaces. [3]
Material
Thermal
Conductivity
- k -
W/(m K)
Temp.
(oC)
Air, atmosphere
(gas)
0.024 25
Aluminum 205 25
Aluminum Brass 121 25
Beef, lean
(78.9% moisture)
0.43 - 0.48 25
Brass 109 25
GM280
(Si, thermal
paste)
1.2 -45~200
Stainless Steel 16 25
Water 0.58 25
Figure 12. Thermal conductivities of various materials. [6][7]

13. Equipment

14. Equipment

15. Procedure
 Measure the temperature distribution for
the steady state conduction of energy
through a uniform flat wall and
demonstrate the effect of a change in the
heat flow.
 Understand the use of the Fourier
Frequency equation to determine the rate
of heat flux through solid materials for
one-dimensional heat flow.
 Measure the temperature distribution for
steady-state energy conduction through a
composite flat wall and determine the
global heat transfer coefficient for a heat
flow through a combination of different
materials in series.
 Determine the thermal conductivity (k) of
a sample of metal.

16. Procedure
 Show that the temperature gradient is
inversely proportional to the cross-
sectional area for one-dimensional heat
flow in a solid material of constant
thermal conductivity.
 Demonstrate the effect of contact
resistance on thermal conduction
between adjacent materials.
 Understand the application of poor
conductors (insulators) and determine
the thermal conductivity k (the
proportionality constant) of an
insulation.
 Observe the conduction of the heat in
an unstable state (qualitative only with a
graphic recorder or a connected PC).

17. Data
Heater
Voltage
(V)
Heater
current
(Amps)
Heated
Section
Temperature
(°C)
Cooled Section
Temperature
(°C)
Cooling
Water
Flowrate
(L/min)
Heat
Flow
(Watts)
V I T1, T2, T3 T6, T7, T8 Fw Q=VI
• As the electrical supply to the heater is Direct Current the power supplied to the
heater is simply obtained from the product of the Voltage and Current, i.e. [5]
Heater Power (Q) = Voltage (V) x Current (I)

18. Calculations
 Steady-State Heat Conduction:
𝑄 = 𝐶
Δ𝑇
Δ𝑥
 The Fourier Rate Equation:
Q = kA
∆T
∆x
, where ∆T = (Ta − Tb)
𝐴 =
π 𝐷2
4
(m2 , cross sectional area of
bar)
 The Overall Heat Transfer
Coefficient:
𝑈 =
𝑄
𝐴(𝑇1 − 𝑇8)
𝐴 =
π 𝐷2
4
(m2 ) cross sectional area
∆𝑇18 = 𝑇1 − 𝑇8 (∘C)
Temperature difference across composite wall

19. Calculations
 Thermal Conductivity: (Constant of
Proportionality)
 Inverse Proportionality of
Temperature Gradient to Area:
∆T =
𝑄∆x
𝑘𝐴
 Effect of Contact Resistance on
Thermal Conduction:

20. Calculations
 Thermal Conductivity and
Application of Insulators:
𝑘 =
𝑄
𝐴 (𝑇ℎ𝑜𝑡𝑓𝑎𝑐𝑒 − 𝑇𝑐𝑜𝑙𝑑𝑓𝑎𝑐𝑒)
 Unsteady State Conduction of
Heat:

21. Safety

22. References
[1] http://philschatz.com/physics-book/contents/m42228.html
[2] Dr. Balku, Ş. (2015). STEADY HEAT TRANSFER AND THERMAL RESISTANCE
NETWORKS. Retrieve from http://slideplayer.com/slide/6638492/
[3] J P Holman, S. B. (2011). HEAT TRANSFER (In SI Units). (10). (M.-H. G. Holdings,
Ed.) New York, United States.
[4] http://www.engr.iupui.edu/~mrnalim/me314lab/lab02.htm
[5] Manual Lab.
[6] http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/thrcn.html
[7] http://www.engineeringtoolbox.com/thermal-conductivity-d_429.html