metal block

8/14/2019 Metal Block

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Jitender Kumar10001033 Experiment No - 12

First order dynamics (Metal block)

AIM: -

To determine the time constant of a first order system in metal block from its response to a step

change in the input variable.

APPARATUS:-

A brass cylindrical metal block. Oil bath with water heater. Digital Temperature indicators to indicate inlet and outlet temperature of cooling

water with accuracy of 0.10C

Stop watch.

Cotton to wipeout oil from the thermometer. An air blower attached at the bottom of the glass tube

PROCEDURE:-

Fill the cylinder vessel with the desired quantity of selected liquid Start the agitator motor and adjust its speed (R.P.M) at desired value. Dip the given Metal block in the oil-bath and allow the temperature to rise around

1200C

Take out the metal block from the oil bath and wipeout oil from its surface withcotton.

Adjust the block at the top of the glass tube and start the air blower. Make sure the flow of air remains constant during the entire period. Note down the temperature as u start the temperature. Repeat above procedure for different speed of air blower.

THEORY:-

Consider a metal block located in a flowing stream of fluid, whose temperature x varies with

time. Our problem is to find out the variation in the metal block reading y with respect to time.

The following assumption will be made to analyze this problem.

All the resistance to heat transfer resides in the film surrounding the block The wall of a block doesnt expand or contract during transient response.

The unsteady-state energy balance equation for this system can be written as

Energy input - Energy output = Accumulation of energy. [1]


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When the block is subjected to some change in the surrounding temperature x(t) either the

thermal energy will enter or leave the block and above equation can be written as

( ) 0 = / [2]

Where,

A= Surface area of the block available for the heat transfer. [m2 ]

Cp= Specific heat of stainless steel [Kcal/Kg*0C]

m= Mass of block [Kg]

t= Time [hr]

h= Film heat transfer coefficient between surrounding fluid and block

[Kcal/hr*m2*0C]

Above equation is a first order different equation. Before solving this equation by means of

Laplace transform, another variable deviation variable shall be introduced. The reason for this

is that as long as the system is at its steady state, it is inconsequential to find indicated

temperature as it is at its desired value. One is interested only when it deviates from its desired

value. In the case of steady-state condition the things do not change with time and equation can

be written as,

( ) 0 = 0 [3]

The subscripts is used to indicate that variable is at steady-state value. Equation 3 states that xs,

ysin other words the thermometer reads true fluid temperature. Subtracting equation (3) from

(2) gives

[ ( ) ( )] 0 = ( )/ [4]

If we define the deviation variable to be difference between variables and its corresponding

steady state value and denote it by in capital then we can write

= ( ) = ( ) [5]

Substituting equation [5] in [4] and dividing both the sides by h*A gives

= ( )/( ) (/) [6]


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Dimensions of

(

) =[] [/ ] [ 2 /] [1/2]= hr

It has a dimension of time alone. As soon as the thermometer is selected values of m, Cp becomes

well defined. When the system of surrounding fluid is selected value of h also gets fixed.

Thus for given metal block and given system m*Cp/h*A has a constant value and has a dimension

of time. Thus this is known as time constant and denoted by . Thus equation [6]can be written

as

= / [7]

Taking Laplace transform of equation (7) and rearranging it as a ration of Y(S) and X(S) gives

()

() =

()+ [8]

The expression on the right hand side of equation [8] is called transfer function of a system

which the ratio of Laplace transform of response of a system(thermometer reading) to Laplace

transform of input variable(surrounding fluid temperature)

By reviewing steps leading to equation [8] it can be seen that the introduction of deviation

variables prior to taking the Laplace transforms of the differential equation results in a transferfunction that is free of initial condition because the initial values of X and Y are zero. In control

system engineering, we are primarily concerned with the deviation of systems variables from

their steady-state values. The use of deviation is natural as well as convenient.

If a step change of magnitude A is introduced in forcing function(input variable) then it can be

written as

X(S) = A/S [9]

Substituting equation [9] in equation [8] and rearranging we get

() =

(+) [10]

Taking Laplace inverse of the above equation,


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() = (1 /) [11]

This is the response of the thermometer to step change in the input variable. From equation

[11] it can be observed that after sufficiently long time contribution due to term e^- t/ will

become zero and Y(t) will approach asymptotically. Several features of this response are:- Value of Y(t) reaches 63.2% of its ultimate value after one time constant, After the elapsed

time of 2, 3,4 and 5 the percentage response is 86.5, 95, 98, and 99.2% respectively.

From these facts one can consider that the response is essentially completed in 5 time

constants.

If response Y(t) is differentiated with respect to time we get,()/ = (/ ) / [12]

Equation [12] gives slope of response curve at any instant of time. Obviously the slope ofresponse curve at time t=0 is given by (A/ )

Thus by making use of tangent to the response curve at t=0 as well as from 63.2% of the response

the time constant of the system can be determined.


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Observation Table

For low flow rate of air blower

time (s)Temperature (0C)

(sec) error in Tpred(0C) error in Temperature

095

97.2 0

22.4193

683.444 20.09 94.567 0.46

39.2590

617.512 8.50 92.655 0.37

65.2787

585.010 2.79 89.81 0.21

91.8184

566.723 0.42 87.04 0.05

118.9781

553.062 2.82 84.335 0.40

145.9178

538.015 5.47 81.777 0.96

177.5875

537.105 5.63 78.92 1.18

207.3171

526.431 7.50 76.379 1.84

249.5967

514.857 9.54 72.987 2.80

311.5564

532.693 6.40 68.45 2.16

36060

539.652 5.18 65.23 1.92

424.457

538.266 5.42 61.354 2.26

483.4654

543.067 4.58 58.165 2.04

583.6150

581.535 2.18 53.459 1.00

620.4647

526.741 7.45 51.926 3.85

736.5944

552.965 2.84 47.694 1.48

838.4441

553.647 2.72 44.631 1.43

1004.4738

578.100 1.58 40.676 0.79

1269.5236.3

626.855 10.14 36.329 4.40

153597.2

686.933 20.70 33.597 7.45

Average time constant 1= 569.13 sec


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Results and Conclusion:-

The average time constants for metal block system are:

For low speed of air blower = 569.13 Seconds.

For high speed of air blower = 388.34 Seconds.

As result clearly portrays that if we increase the speed of air blower the heat transfer rate willincrease because we are increasing the velocity of fluid i.e. air and we know that heat transfer

coefficient depends on the velocity (heat transfer coefficient depends on Reynolds number and

Reynolds number depends on velocity) so if h increases heat transfer rate will increase it will

take less time to reach in steady state so time constant will decrease.

The difference between thermometer experiment and in this experiment is that this experimentdone in forced convection and the thermometer has been done in natural convection so we

have observed that time constant for forced convection is less as compare to the natural

convection, reason is same velocity of fluid increases.

metal block

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