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

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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:

C_m                - thermal capacitance of mercury, J/K;

C_p^m              - specific heat of mercury, J/(kg*K);

M_m               - mass of mercury, kg;

∆Q^m_accum.       - amount of heat energy accumulated by mercury during a period of time ∆t, J;

∆Q^m_in            - amount of heat energy transferred to mercury during a period of time ∆t, J;

∆Q^m_out           - outflow of heat energy from mercury during a period of time ∆t, J;

R_f,m              - thermal resistance between mercury and outside fluid,K/W;

∆t                 - period of time, s;

T_m                - temperature of mercury, K;

∆T_m/∆t         - rate of change of temperature of mercury, K/s;

dT_m/dt         - instantaneous rate of change of temperature of mercury, K/s;

T_fl                - temperature of the fluid outside the bulb, K.

(10)

A_g                - heat transfer surface area, m²;

h_fl,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.

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 = R_f,mC_m      - time constant, s.  Let:   A=10°C;       

          R_f,m = 131, K/W;               C_m = 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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