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Masters of Science in Energy Engineering
Exercises of Trasmissione del calore
Professor
Student
Vittorio Ferraro
Wahab Uddin(187950)
Academic Year 2016~2017
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Heat and Mass Transfer Report, Exercises of Heat and Mass Transfer
Describes the solution of heat and mass transfer through conduction, convection and boiling in 1D and 2D using the solution of some numerical problems.
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Masters of Science in Energy Engineering
Exercises of Trasmissione del calore
Professor Student
Vittorio Ferraro Wahab Uddin(187950)
Academic Year 2016~
TRANSIENT CONDUCTION IN 1-D
A concrete wall has a thickness of 30 cm at a temperature of 40 ° C. One of the two faces is in
contact with a gas at a temperature of 800 ° C. Assuming that the inner surface of the wall is
insulated, evaluate the time after which the temperature at the adiabatic side is equal to 600 ° C.
Determine also the distribution of the wall temperatures in the same time instant and the
amount of heat transmitted per unit of surface.
DATA
K 0,8 kcal/(hmK) H 25 kcal/(hm²K) To 40 °C L 0,3 M Ρ 2300 kg/m³ C 0,2 kcal/(kgK) T∞ 800 °C
(𝑇(0,𝑡)−𝑇∞)/( T 𝑜−𝑇∞)=0,
Bi hL/k 9, 1/Bi 0, Fo By Diagram 0, t=FoρcL²/k 41,4 h If x/L = 1 (𝑇(x,𝑡)−𝑇∞)/( T (0, t)−𝑇∞)=0,
Bi²Fo = 70
Q / Qmax = 0.7 by diagram
Q = - 146 832 kcal / m²
Transient Conduction 2-D
In the design of a fire extinguishing system must be assessed at the time the limit below which
the wooden beams bear the load of the fire. The beams are long with section 100 * 50 mm ^ 2
and are located at the 16 ° C temperature. When the fire broke out, the beams are exposed to hot
gases at a temperature of 540 ° C. Estimate the time it takes for the wood to reach the limit value
of combustion temperature estimated at 480 ° C.
DATA
T(x,y,0) 16 °C T∞ 540 °C Tcomb 480 °C H 15 K 0, Α 0, Ρ 800 C 0, L 0,025 M L 0,05 M
Bix 1,293 1/Bix 0, Biy 2,586 1/Biy 0,
We see for a time equal to 30 minutes.
Hence the result is different from 0.115.
Let us for a time of 40 minutes.
Conduction in Semi-infinite Wall
Tracing the temperature profile inside a semi-infinite wall consisting of lead with density ρ =
11190 kg / m ^ 3, thermal conductivity k = 30 kcal/hmk and heat capacity c = 0.03 kcal / kgK.
The wall is initially at a temperature of 20 ° C, and is placed in contact with a source temperature
of 200 ° C in correspondence to 3 instants of time: T1 = 15 min; T2 = 1h; T3 = 2h.
Determine the temperature profiles derived from differential and integral analysis. Evaluate the
error percentage and standard deviation.
DATA
Ρ 11190 kg/m K 30 kcal/hmK C 0,03 kcal/kgK To 20 °C Ts 200 °C α 0,089366 m²/h
Differential
formulation:
Integral
formulation:
DIFFERENTIAL
METHOD
t= 0,25 δ 1, X 0 0,1 0,2 0,3 0,4 0,5 0, T(x,t) 200 134,5087 81,93297 48,05001 30,52118 23,242126 20, X 0,7 0,8 0,9 1 1, T(x,t) 20,16702 20,02771 20,0037 20,0004 20 t= 1 δ 2, X 0 0,1 0,2 0,3 0,4 0,5 0, T(x,t) 200 166,3428 134,5087 106,03 81,93297 62,6479352 48, X 0,7 0,8 0,9 1 1,1 1,2 1, T(x,t) 37,59858 30,52118 25,98818 23,24213 21,6687 20,816 20, X 1,4 1,5 1,6 1,7 1,8 1,9 2 T(x,t) 20,16702 20,06985 20,02771 20,01042 20,00372 20,00126 20, X 2,1 2, T(x,t) 20,00012 20, t= 0,25 δ 304391451 x 0 0,1 0,2 0,3 0,4 0,5 0, T(x,t) 200 176,0902 152,8384 130,849 110,6258 92,5388809 76,
INTEGRAL
Forced Convection
A flow of air at a temperature of 20 ° C and 1 atm pressure, flowing on a plate at a speed of 5 m
/ s. Assuming that the temperature of the panel is 56 ° C, and that the width of the plate is 0.5 m,
evaluate at abscissa x = 5 m the hydrodynamic boundary layer, the thermal boundary layer, the
coefficient of friction local and the mean, average and local stresses due to friction, the
coefficient of convective heat exchange and the local mean, the thermal power exchanged by the
plate, the performance of the long x and y speed, the temperature profile at x = 1.5 m, the trend
of the hydrodynamic boundary layer up to x = 2Xcr and, relatively to the turbulent part, compare
the results obtained with the exact formula with those of the approximate formula.
DATA
To 56 °C T∞ 20 °C V∞ 5 m/s X 0,5 M Recr 500000 Tfilm 38 °C Ρ 1,136 kg/m³ Μ 1,91E- 05 Pa·s Cp 1004,8 J/kgK Ν 1,68E- 05 m²/s K 0,027 W/mK Pr 0, α (^) 0,0841 m²/s
For x=0,5 m; Rex= 148691,1.
= 6,48 mm
= 7,28 mm
See the table with the velocity and temperatures:
y η f Df d²f Vx Vy y η* df 0,001 0,8 0,10611 0,26471 0,32739 1,3 0,00064 0,001 0,7 0, 0,002 1,5 0,37165 0,486515 0,30352 2,4 0,002322 0,002 1,4 0, 0,003 2,3 0,85175 0,705155 0,23822 3,5 0,005055 0,003 2,1 0, 0,004 3,1 1,482965 0,86107 0,150245 4,3 0,007607 0,004 2,8 0, 0,005 3,9 2,210 905 0,94832 0,072185 4,7 0,009374 0,005 3,5 0, 0,006 4,6 2,88 826 0,98269 0,02948 4,9 0,010755 0,006 4,1 0, 0,006483 5,0 3 ,28 329 0,99155 0,01591 5,0 0,010856 0,00724 5,0 0,
𝐶𝑓𝑥=0,664 / √𝑅𝑒𝑥 = 1,72197E- 03
𝜏𝑝=𝐶𝑓𝑥1/2^ 𝜌 𝑣∞^2 = 2,44520E- 02
= 6,196101 W/m^2K
= 12,3922 W/m^2K
= 111,5298 W
For x=1,5 m Re 446073, δi 0,011229 M Δt 0,012529 M y η f df Vx vy y η* df T 0,001 0,4 0,0340977 0,147802 0,74 0,000119 0,001 0,4 0,13277 51, 0,002 0,9 0,1331259 0,294279 1,47 0,000483 0,002 0,8 0,26471 46, 0,003 1,3 0,2965821 0,436368 2,18 0,001072 0,003 1,2 0,39378 41, 0,004 1,8 0,5195509 0,569474 2,85 0,001852 0,004 1,6 0,51676 37, 0,005 2,2 0,8003734 0,687797 3,44 0,002736 0,005 2,0 0,62977 33,
Vy Trend
X δix Rex δixapp
- X 0,7 0,8 0,9 1 1,1 1,2 1,
- T(x,t) 63,50218 52,55794 43,80373 36,9940 31,84277 28,05326 25,
- X 1,4 1,5 1,6 1,7 1,8 1,9
- T(x,t) 23,45621 22,18019 21,34065 20,8034 20,46927 20,26703 20,
- X 2,1 2,2 2,3 2,4 2,5 2,6 2,
- T(x,t) 20,07993 20,04204 20,02153 20,0107 20,00521 20,00247 20,
- X 2,8 2,9
- T(x,t) 20,00050 20,00022 20,
- t= 0,25 δ 0, METHOD
- X 0 0,1 0,2 0,3 0,4 0,5 0,
- T(x,t) 200 137,1865 87,8009 51,84335 29,31381
- X 0,7 0,8 0,9 1 1,1 1,2 1,
- T(x,t)
- X 1,4 1,
- T(x,t)
- t= 1 δ
- X 0 0,1 0,2 0,3 0,4 0,5 0,
- T(x,t) 200 166,9147 137,1865 110,8152 87,8009 68,1436288 51,
- X 0,7 0,8 0,9 1 1,1 1,2 1,
- T(x,t) 38,9000 29,31381 23,08453
- X 1,4 1,
- T(x,t)
- t= 2 δ 1,
- X 0 0,1 0,2 0,3 0,4 0,5 0,
- 0,006 2,7 1,1300545 0,744772 3,72 0,003218 0,006 2,4 0,72899 29,
- 0,007 3,1 1,4985123 0,863781 4,32 0,004468 0,007 2,8 0,81152 26,
- 0,008 3,6 1,8962038 0,771501 3,86 0,003189 0,008 3,2 0,87609 24,
- 0,009 4,0 2,3143353 0,95603 4,78 0,005678 0,009 3,6 0,92333 22,
- 0,01 4,5 2,7456143 0,977723 4,89 0,006018 0,01 4,0 0,95552 21,
- 0,011229 5,0 3,2852668 0,991577 4,96 0,006261 0,011 4,4 0,97587 20, - 0,012 4,8 0,97928 20, - 0,0125 5,0 0,99155
- 0,1 0,002899 29738,22 0,
- 0,2 0,0041 59476,44 0,
- 0,3 0,005022 89214,66 0,
- 0,4 0,005799 118952,9 0,
- 0,5 0,006483 148691,1 0,
- 0,6 0,007102 178429,3 0,
- 0,7 0,007671 208167,5 0,
- 0,8 0,008201 237905,8 0,
- 0,9 0,008698 267644 0,
- 1 0,009169 297382,2 0,
- 1,1 0,009616 327120,4 0,
- 1,2 0,010044 356858,6 0,
- 1,3 0,010454 386596,9 0,
- 1,4 0,010849 416335,1 0,
- 1,5 0,011229 446073,3 0,
- 1,6 0,011598 475811,5 0,
- 1,7 0,012445 505549,7 0,
- 1,8 0,01533 535288 0,
- 1,9 0,018085 565026,2 0,
- 2 0,020737 594764,4 0,
- 2,1 0,023308 624502,6 0,
- 2,2 0,025809 654240,8 0,
- 2,3 0,028251 683979,1 0,
- 2,4 0,030641 713717,3 0,
- 2,5 0,032985 743455,5 0,
- 2,6 0,035289 773193,7 0,
- 2,7 0,037555 802931,9 0,
- 2,8 0,039788 832670,2 0,
- 2,9 0,04199 862408,4 0,
- 3 0,044163 892146,6 0,
- 3,1 0,04631 921884,8 0,
- 3,2 0,048432 951623 0,
- 3,3 0,050531 981361,3 0,
- 3,36 0,05178 999204,2 0,