In the production of sheet metals or plastics, it is custom- ary to cool the material before it leaves the production process for storage or shipment to the customer. Typi- cally, the process is continuous, with a sheet of thickness 8 and width W cooled as it transits the distance L between two rollers at a velocity V. In this problem, we consider cooling of plain carbon steel by an airstream moving at a velocity u. in cross flow over the top and bottom surfaces of the sheet. A turbulence promoter is used to provide tur- bulent boundary layer development over the entire surface. Ti 8 Turbulence romoter W + Surroundings, Tsur Air uoo X Too Plain carbon steel L Vv (a) By applying conservation of energy to a differential control surface of length dx, which either moves with the sheet or is stationary and through which the sheet passes, and assuming a uniform sheet temperature in the direction of airflow, derive a dif- ferential equation that governs the temperature dis- tribution, T(x), along the sheet. Consider the effects of radiation, as well as convection, and express your result in terms of the velocity, thickness, and properties of the sheet (V, S, p, C., &), the average convection coefficient hw associated with the cross flow, and the environmental temperatures (T., Tsur). (b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For 8 = 3 mm, V = 0.10 m/s, L = 10 m, W = 1 m, u = 20 m/s, T = 20°C, and a sheet temperature of T, = 500°C at the onset of cooling, what is the outlet temperature T? Assume a negligible effect of the sheet velocity on bound- ary layer development in the direction of airflow. The density and specific heat of the steel are p = 7850 kg/m³ and = 620 J/kg K, while properties Cp of the air may be taken to be k 0.044 W/m K, v=4.5 × 105 m²/s, Pr = 0.68. . =
In the production of sheet metals or plastics, it is custom- ary to cool the material before it leaves the production process for storage or shipment to the customer. Typi- cally, the process is continuous, with a sheet of thickness 8 and width W cooled as it transits the distance L between two rollers at a velocity V. In this problem, we consider cooling of plain carbon steel by an airstream moving at a velocity u. in cross flow over the top and bottom surfaces of the sheet. A turbulence promoter is used to provide tur- bulent boundary layer development over the entire surface. Ti 8 Turbulence romoter W + Surroundings, Tsur Air uoo X Too Plain carbon steel L Vv (a) By applying conservation of energy to a differential control surface of length dx, which either moves with the sheet or is stationary and through which the sheet passes, and assuming a uniform sheet temperature in the direction of airflow, derive a dif- ferential equation that governs the temperature dis- tribution, T(x), along the sheet. Consider the effects of radiation, as well as convection, and express your result in terms of the velocity, thickness, and properties of the sheet (V, S, p, C., &), the average convection coefficient hw associated with the cross flow, and the environmental temperatures (T., Tsur). (b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For 8 = 3 mm, V = 0.10 m/s, L = 10 m, W = 1 m, u = 20 m/s, T = 20°C, and a sheet temperature of T, = 500°C at the onset of cooling, what is the outlet temperature T? Assume a negligible effect of the sheet velocity on bound- ary layer development in the direction of airflow. The density and specific heat of the steel are p = 7850 kg/m³ and = 620 J/kg K, while properties Cp of the air may be taken to be k 0.044 W/m K, v=4.5 × 105 m²/s, Pr = 0.68. . =
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Please show all steps not Ai generated I need to understand the process
Transcribed Image Text:In the production of sheet metals or plastics, it is custom-
ary to cool the material before it leaves the production
process for storage or shipment to the customer. Typi-
cally, the process is continuous, with a sheet of thickness
8 and width W cooled as it transits the distance L between
two rollers at a velocity V. In this problem, we consider
cooling of plain carbon steel by an airstream moving at a
velocity u. in cross flow over the top and bottom surfaces
of the sheet. A turbulence promoter is used to provide tur-
bulent boundary layer development over the entire surface.
Ti
8
Turbulence
promoter
-Surroundings, Ts
sur
W
+
Air
U∞o
Too
Plain carbon steel
L
V
(a) By applying conservation of energy to a differential
control surface of length dx, which either moves
with the sheet or is stationary and through which
the sheet passes, and assuming a uniform sheet
temperature in the direction of airflow, derive a dif-
ferential equation that governs the temperature dis-
tribution, T(x), along the sheet. Consider the effects
of radiation, as well as convection, and express
your result in terms of the velocity, thickness, and
properties of the sheet (V, S, p, Cp, E), the average
convection coefficient hw associated with the cross
flow, and the environmental temperatures (T., Tsur).
(b) Neglecting radiation, obtain a closed form solution to
the foregoing equation. For d = 3 mm, V = 0.10 m/s,
L = 10 m, W = 1 m, u = 20 m/s, T = 20°C, and
a sheet temperature of T; = 500°C at the onset of
cooling, what is the outlet temperature To? Assume
a negligible effect of the sheet velocity on bound-
ary layer development in the direction of airflow.
The density and specific heat of the steel are p =
7850 kg/m³ and cp 620 J/kg K, while properties
of the air may be taken to be k = 0.044 W/m. K,
v=4.5 x 10-5 m²/s, Pr = 0.68.
=
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