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:

• 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.

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

c^g_p               - specific heat of glass, J/(kg*K);

M_g                - mass of glass bulb, kg;

∆Q^g_accum        - amount of heat energy accumulated by glass bulb during a period of time ∆t,J;

∆Q^g_in            - amount of heat energy transferred to glass bulb from fluid during a period of time ∆t, J;

∆Q^g_out           - outflow of heat energy from glass bulb during a period of time ∆t, J;

R_f,g               - thermal resistance of fluid film and glass wall, K/W;

R_g,m              - thermal resistance of glass and mercury film, K/W;

T_g                - temperature of glass, K.

(37)

(38)

where,

A_g                - heat transfer surface area, m²;

h_f,h_m            - film coefficients of fluid and mercury, respectively, W/(m²*K);

k_g                 - thermal conductivity of glass, W/(m*K);

x_g                 - thickness of glass wall, m.

Transfer function is as follows:

(39)

where,                   τ_1,2 = R_fl,gC_m..

Let,    τ₁ τ₂ =τ²,       (39)          and    τ₁₊ τ₂₊ τ_1,2= 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= 10^oC 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