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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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