UNIVERSITY OF TORONTO
DEPARTMENT OF MECHANICAL & INDUSTRIAL ENGINEERING
MIE313H-Heat and Mass Transfer
Laboratory Experiment ⅠV
Winter 2023
NOTES:
(i) Attempt all 4 parts of the project
(ii) Total mark: 100 points
Name:
Surname:
Student #:
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Objective:
1. To study the natural convection mechanism for cooling processes.
2. To analyze the multimode heat transfer.
Equipment:
•
Energy2D simulator.
Part A) Natural Convection:
One of the fundamental problems of interest in convective heat transfer is cooling processes with
natural convection. While the geometry could not be any more basic, as shown in Figure 1, occurs
in a number of engineering applications. In this flow type, the boundary layers develop freely,
without any constraints imposed by adjacent surfaces. Consequently, there will always be a
region of the flow outside the boundary layer in which velocity and temperature gradients are
negligible.
As shown in Figure 1, the heat generated inside an aluminum wall (height=0.5 m, width=0.04 m,
depth= 1m, and 𝑞̇ = 200 𝑊⁄ 3 ) will be cool down by surrounding gas. Insulations have
𝑚
surrounded the wall. Therefore, it can only transfer energy to the surrounding gas. The
temperature of the surface of the wall will be measured during the process. The initial temperature
of the aluminum wall is 𝑇𝑖 = 500𝑜 𝐶 .
Page 1 of 4
Figure 1: Cooling of the wall with natural convection
In this part, the wall is only cooled down by the natural convection mechanism with different
conditions, as shown in the following Table:
Gas’s
Thermal
Kinematic
Thermal
Thermal
Temperature (C)
Expansion
Viscosity
Diffusivity
Conductivity
(m2/s)
(m2/s)
(W/m.C)
Coefficient (1/C)
Case 1
50
0.000005
0.002
0.0001
0.02
Case 2
100
0.000002
0.0025
0.00024
0.05
Case 3
150
0.000001
0.003
0.00036
0.07
Run the “part1_case1.e2d”, “part1_case2.e2d”, and “part1_case3.e2d” files and stop the
simulation once it becomes a steady state.
Page 2 of 4
1-Find the average convective heat transfer coefficient (ℎ̅) of different cases using the following
equation (15 Points):
𝑞" = ℎ̅(𝑇𝑠 − 𝑇∞ )
(1)
where
𝑞" = ℎ𝑒𝑎𝑡 𝑓𝑙𝑢𝑥, 𝑇∞ = 𝑔𝑎𝑠’𝑠 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 ,
𝑇𝑠 = 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑎𝑙𝑙
2- Find the coefficients a and b for the following correlation expressing the relationship between
̅̅̅̅̅̅𝐿 ) and the Rayleigh number (𝑅𝑎𝐿 ) in a vertical cavity (35 Points):
the Nusselt number (𝑁𝑢
̅̅̅̅̅̅
𝑁𝑢𝐿 = 𝑎(𝑅𝑎𝐿 )𝑏
(2)
where
̅
ℎ𝐿
̅̅̅̅̅̅
𝑁𝑢𝐿 =
,
𝑘
𝑔𝛽(𝑇𝑠 − 𝑇∞ )𝐿3
𝑅𝑎𝐿 =
𝑣𝛼
𝛽 = 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝐸𝑥𝑝𝑎𝑛𝑠𝑖𝑜𝑛 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 , 𝑣 = 𝐾𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦
𝛼 = 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝐷𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 =
𝑘
𝜌𝐶𝑝
𝑔 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 = 9.81 𝑚/𝑠 2 , 𝐿 = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑤𝑎𝑙𝑙
Hint: By taking the natural log, equation 2 will turn into a linear equation:
̅̅̅̅̅̅𝐿 ) = 𝐿𝑛(𝑎) + 𝑏 × 𝑙𝑛(𝑅𝑎𝐿 )
𝑙𝑛(𝑁𝑢
Then, by determining Nusselt and Rayleigh numbers of each case, fit a linear curve and find a
and b.
(For Fitting you can use any software like excel, origin, etc. Also, a handout has been
uploaded on Quercus which shows how to fit a curve using Python)
Page 3 of 4
Part B) Multimode Heat Transfer:
In this part, the same setup (an aluminum wall with of height=0.5 m, width=0.04 m, depth= 1m,
and 𝑞̇ = 200 𝑊⁄ 3) will be used to analysis the effect of radiation in the cooling process. It should
𝑚
be mentioned that the initial temperature of the aluminum wall is 𝑇𝑖 = 500𝑜 𝐶, and temperature of
gas is 𝑇∞ = 0𝑜 𝐶 .
Consider the following scenarios in the cooling process of the wall:
•
The wall is only cooled down by natural convection mechanism (part2_single_mode.e2d).
•
In addition to natural convection, the wall can transfer energy by radiation to the
surrounding walls (𝑇𝑤𝑎𝑙𝑙𝑠 = 0𝑜 𝐶), (part2_multi_mode.e2d).
3- Derive the expressions for the temperature of the wall for both cases. Write down all
assumptions of the expressions and initial/boundary conditions. (You may use the results of the
simulations for some of your assumption) (25 Points)
4- Run the “part2_single_mode.e2d” and “part2_multi_mode.e2d” files and stop the simulation
after 45 minutes of CFD time (CFD time is shown on the right corner of each simulation). Compare
the temperature of the wall in both cases. What is the effect of radiation on the temperature of the
wall and the rate of heat transfer? (25 Points)
Page 4 of 4