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,
ฯ = 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