Why bulb of mercury thermometer is kept cylindrical ?

We all know that mercury thermometer is used for measurement of temperature. And our prime intention is to get accurate temperature measurement.

Mercury thermometer

Principle of Temperature measurement in mercury thermometer is thermal expansion. If temperature increases mercury expands and if it decreases mercury contracts. So basically temperature difference causes the transfer of heat and there is the process leading towards the thermal equilibrium. And the duration in which process of reaching the equilibrium is there is time dependent heat transfer - very well know as transient heat transfer.

For the analysis of transient heat transfer there is one method- lumped mass system. Based on it following equation is obtained.

 Equation 1

Where,

T              = Temperature of body at time 't'

Tinf          = Temperature of surrounding of the body,

Ti             = Initial Temperature of body 

t               = time(seconds)

h              = convective heat transfer coeffienct

A             = surface area of the body 

Rho         = density of the body material

c              = specific heat of the body

V             = volume of the body 

   Here, ( Rho*c*V/(h*A) )= time constant =Tau

So from Equation 1, we can see that to reach the temperature Tinf , we should have theoretically infinite time. But to reach specific temperature, Tau plays a major role. If Tau(=time constant) is less then time required to reach the specific temperature is less, if it is high then so time is also high. 

Graph 1

Now we can understand that to measure the temperature we have to wait for infinite time(see graph 1), which is impossible. So we should measure the temperature as near as actual temperature with possible time.  i.e. we should have as minimum as possible the time constant(refer graph 2). For given material with same convective heat transfer coefficient,  Spherical shape has minimum ratio of A/V . Here we want high A/V, to make (Rho*c/h) *(V/A) minimum.  The cylindrical one has high (A/V).

Graph 2

Thus we should have minimum time constant to measure the temperature by thermometer. And for that we should have cylindrical bulb. That's why bulb of mercury thermometer is kept cylindrical. hence more sensitive.  

See above video for: How to plot transient temperature in scilab and view the temperature variation.