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:
Its better you read the previous two post before going through this, for better understanding:
- Liquid-in-glass thermometer : Introduction
- Dynamic characteristic of liquid-in-glass thermometer (thermal capacitance of glass wall not included)
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 ∆t→0, 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, τ1τ2 =τ2, (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
