Using Numerical Method In Engineering Application
HEAT TRANSFER OF BALL
BEARING
Muhammad Sabry Anwar Bin
Sazali
2011297756
BACKGROUND
A steel ball bearing at 1200K is need to be quenched for 30 seconds in water that is maintained at a room temperature of 300K after heating them to a high temperature of 1000K. However, it takes time to take the ball from the furnace to the quenching bath and its temperature falls. If the temperature of the furnace is 1200K and it takes 10 seconds to take the ball to the quenching bath, is less time needed or more time needed to reach temperature of 1000K
I assumed assumed that the ball bearing is cooled by radiation and convection to its surroundings.
I decided to use 4th order Runge Kutta method to solve the problem.
Objective
To solve the first order nonlinear ordinary differential equations using numerical method. To understand the 4th order Runge Kutta method and its applications.
Problem statement
Steel ball bearing radius 0.02m, ρ = dT 7800kg/m3
A T 4 Ta4 mC dt The radiation equation is
The convection equation is
Assumed T0= 1200K and Ambient temperature, Assumed that all heat transfer in radiation and convection only.
Mathematical model
Combining both convection and radiation equation to form a new rate of heat lost equation:
After subsitution of the constants, the equation reduced to:
Preliminary Solution
Rewrite the equations:
=(((-2.20673*(10^-13))*(T^4))-((1.60256*(10^-2))*(T))
+(4.8095))
f(t,T)=(((-2.20673*(10^-13))*(T^4))-((1.60256*(10^-2))*(T))
+(4.8095))
Set
Runge Kutta 4th Order Solution
1
Ti 1 Ti k1 2k 2 2k3 k 4 h
6
k1 f (ti , Ti )
1
1 k 2 f (ti h, Ti k1h)
2
2
1
1 k3 f (ti h, Ti k 2 h)
2
2 k 4 f (ti h, Ti k3 h)
MATLAB IMPLEMENTATION
Result
Discussion and conclusion
The steel ball bearing will reach temperature
1000K at 15.3 seconds
The time taken for the ball bearing to cool from
1200K to 1000K is obtained by using the 4 th order Runge Kutta method
BEARING
Muhammad Sabry Anwar Bin
Sazali
2011297756
BACKGROUND
A steel ball bearing at 1200K is need to be quenched for 30 seconds in water that is maintained at a room temperature of 300K after heating them to a high temperature of 1000K. However, it takes time to take the ball from the furnace to the quenching bath and its temperature falls. If the temperature of the furnace is 1200K and it takes 10 seconds to take the ball to the quenching bath, is less time needed or more time needed to reach temperature of 1000K
I assumed assumed that the ball bearing is cooled by radiation and convection to its surroundings.
I decided to use 4th order Runge Kutta method to solve the problem.
Objective
To solve the first order nonlinear ordinary differential equations using numerical method. To understand the 4th order Runge Kutta method and its applications.
Problem statement
Steel ball bearing radius 0.02m, ρ = dT 7800kg/m3
A T 4 Ta4 mC dt The radiation equation is
The convection equation is
Assumed T0= 1200K and Ambient temperature, Assumed that all heat transfer in radiation and convection only.
Mathematical model
Combining both convection and radiation equation to form a new rate of heat lost equation:
After subsitution of the constants, the equation reduced to:
Preliminary Solution
Rewrite the equations:
=(((-2.20673*(10^-13))*(T^4))-((1.60256*(10^-2))*(T))
+(4.8095))
f(t,T)=(((-2.20673*(10^-13))*(T^4))-((1.60256*(10^-2))*(T))
+(4.8095))
Set
Runge Kutta 4th Order Solution
1
Ti 1 Ti k1 2k 2 2k3 k 4 h
6
k1 f (ti , Ti )
1
1 k 2 f (ti h, Ti k1h)
2
2
1
1 k3 f (ti h, Ti k 2 h)
2
2 k 4 f (ti h, Ti k3 h)
MATLAB IMPLEMENTATION
Result
Discussion and conclusion
The steel ball bearing will reach temperature
1000K at 15.3 seconds
The time taken for the ball bearing to cool from
1200K to 1000K is obtained by using the 4 th order Runge Kutta method