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Monday, July 16, 2012

Dynamic characteristic of liquid-in-glass thermometer (thermal capacitance of glass wall is included)





Its a derivation  part which is a continuation of the previous two post. Here the dynamic characteristics of liquid-in-glass thermometer by considering the thermal capacitance of glass wall is discussed.

Its better you read the previous two post before going through this, for better understanding:





Dynamic characteristic of liquid-in-glass thermometer (thermal capacitance of glass wall is included)

The heat energy balance:

(20)
Thermal capacitance of the glass walls is included.

a). The heat energy balance for mercury in the bulb:


Figure 1:. Liquid-in-glass thermometer.

(25)(26)(27)
We get the first order differential equation:

(28)

b). The heat energy balance for the glass wall has both inflow and outflow of heat:

(29)(30)
(31)(32)
(33)
Let ∆t0, then
(34)
Substitute (28) into (34) and after manipulations we get:

(35)
or
(36)
Equation (36) is a second-order differential equation.

Cg                - thermal capacitance of glass bulb, J/K;

cgp               - specific heat of glass, J/(kg*K);
Mg                - mass of glass bulb, kg;
∆Qgaccum        - amount of heat energy accumulated by glass bulb during a period of time ∆t,J;
∆Qgin            - amount of heat energy transferred to glass bulb from fluid during a period of time ∆t, J;
∆Qgout           - outflow of heat energy from glass bulb during a period of time ∆t, J;
Rf,g               - thermal resistance of fluid film and glass wall, K/W;
Rg,m              - thermal resistance of glass and mercury film, K/W;
Tg                - temperature of glass, K.


(37)
(38)

where,
Ag                - heat transfer surface area, m2;
hf,hm            - film coefficients of fluid and mercury, respectively, W/(m2*K);
kg                 - thermal conductivity of glass, W/(m*K);
xg                 - thickness of glass wall, m.

Transfer function is as follows:

(39)
where,                   τ1,2 = Rfl,gCm..
Let,    τ1 τ22,       (39)          and    τ1+ τ2+ τ1,2= 2 ζτ.   (40)

Then, 

(41)
Let:    T'fl = A - step change, and ζ = 1. Then we have:

(42)
Use inverse Laplace transform:

(43)
Let:    A= 10oC and τ = 85,s. Then we can plot a transient response of this thermometer to a step change in the input variable (see Figure 2).

Figure 2:. Transient response of liquid-in-glass thermometer.

Article Source:: Dr. Alexander Badalyan, University of South Australia

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Dynamic characteristic of liquid-in-glass thermometer (thermal capacitance of glass wall is not included)






This post describes the dynamic characteristics of liquid-in-glass thermometer which contains a derivation part using the heat energy balance equation.Its the continuation of the previous post titled 'Liquid-in-glass Thermometer'. 

Dynamic characteristic of liquid-in-glass thermometer (thermal capacitance of glass wall is not included):

The heat energy balance for mercury in the bulb:

(1)

Thermal capacitance of the glass walls is neglected.


(8)(6)(7)

Let Dt®0, then we can get a first order differential equation:
(9)

Explanations of variables used in the above equations is given below:

Cm                - thermal capacitance of mercury, J/K;
Cpm              - specific heat of mercury, J/(kg*K);
Mm               - mass of mercury, kg;
∆Qmaccum.       - amount of heat energy accumulated by mercury during a period of time ∆t, J;
∆Qmin            - amount of heat energy transferred to mercury during a period of time ∆t, J;
∆Qmout           - outflow of heat energy from mercury during a period of time ∆t, J;
Rf,m              - thermal resistance between mercury and outside fluid,K/W;
∆t                 - period of time, s;
Tm                - temperature of mercury, K;
∆Tm/∆t         - rate of change of temperature of mercury, K/s;
dTm/dt         - instantaneous rate of change of temperature of mercury, K/s;
Tfl                - temperature of the fluid outside the bulb, K.

(10)

Ag                - heat transfer surface area, m2;
hfl,hm            - film coefficients of fluid and mercury, respectively, W/(m2*K);
kg                 - thermal conductivity of glass, W/(m*K);
xg                 - thickness of glass wall, m.

Differential equation with variables in deviation form:

(11)(12)

Let:    T'fl = A - step change. Then we have:

(13)(14)
(15)

Use inverse Laplace transform:

(16)
where,


τ = Rf,mCm      - time constant, s.  Let:   A=10°C;       
          Rf,m = 131, K/W;               Cm = 0.56, J/K.

Figure 2 shows a dynamic response of this thermometer to a step change in temperature.

Figure 2.: Dynamic response of liquid-in-glass thermometer to a step change in temperature.
From equation (12) we can get:

(17)(18)
Using a block diagram in Figure 3 we can get the following expression for a transfer function:

Figure 3.: Block diagram of a thermometer.

(19)

Article Source:: Dr. Alexander Badalyan, University of South Australia

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