11-19-2016, 11:33 AM

2D Temperature Distribution in vertical plates. [ TXY ]

From the author’s Engineering Collection, included in the ETSII4 module.

This program calculates the temperature distribution T(x,y) within a rectangular vertical plate with dimensions (b x h); with a known temperature distribution on its upper side - either constant or varying with x - T(x,h), immersed in a uniform ambient temperature T0.

The expression used is based on an infinite sum as follows:

T(x,y) = T0 + 2/b SUM T(n,x) ; n= 1,2,....

with the following general term, where Mn = pi. n / b

Tn(x,y) = sh(Mn.y). sin(Mn.x) / sh (Mn.h) INTG { T(x,b) – T0] sin Mn t} dt ; between [0, b]

The numerical integration is done using the "ITG" routine also included in the module.

Example:

Calculate the temperature in the points P(1, 2) and Q(2, 3) within a flat plate of dimensions (2 x 5) m, with a temperature distribution on its top side given by the function: t(x,5) = x^2 + 10 deg C. The ambient temperature is t0 = 10 deg C. Compare the result with the case of a constant temperature on the top side t(x, 5) = 100 deg C.

The solutions are shown below.

The temperature function for the second case can be easily programmed:

01 LBL “TX”

02 X^2

03 10

04 +

05 END

From the author’s Engineering Collection, included in the ETSII4 module.

This program calculates the temperature distribution T(x,y) within a rectangular vertical plate with dimensions (b x h); with a known temperature distribution on its upper side - either constant or varying with x - T(x,h), immersed in a uniform ambient temperature T0.

The expression used is based on an infinite sum as follows:

T(x,y) = T0 + 2/b SUM T(n,x) ; n= 1,2,....

with the following general term, where Mn = pi. n / b

Tn(x,y) = sh(Mn.y). sin(Mn.x) / sh (Mn.h) INTG { T(x,b) – T0] sin Mn t} dt ; between [0, b]

The numerical integration is done using the "ITG" routine also included in the module.

Example:

Calculate the temperature in the points P(1, 2) and Q(2, 3) within a flat plate of dimensions (2 x 5) m, with a temperature distribution on its top side given by the function: t(x,5) = x^2 + 10 deg C. The ambient temperature is t0 = 10 deg C. Compare the result with the case of a constant temperature on the top side t(x, 5) = 100 deg C.

The solutions are shown below.

Code:

`Point T(x,5)=100 T(x, 5) = x^2 + 10`

-----------------------------------------------------

P(1, 2) T(1, 2) = 10.0239 C T(1, 2) = 10,01357914

Q(2, 3) T(2, 3) = 10.00 C T(2, 3) = 10,00 C

The temperature function for the second case can be easily programmed:

01 LBL “TX”

02 X^2

03 10

04 +

05 END