A sphere, initially at a constant temperature, is immersed in a liquid whose temperature is maintained constant. The time t taken for the temperature of the centre of the sphere to reach a given temperature $\theta_c$ is a function of the following variables:
Diameter of sphere, d
Thermal conductivity of sphere, k
Density of sphere, ρ

Question

A sphere, initially at a constant temperature, is immersed in a liquid whose temperature is maintained constant. The time t taken for the temperature of the centre of the sphere to reach a given temperature [latex] heta_c[/latex] is a function of the following variables:
Diameter of sphere, d
Thermal conductivity of sphere, k
Density of sphere, ρ

Specific heat capacity of sphere, [latex]C_p[/latex]
Temperature of fluid in which it is immersed, [latex] heta_s[/latex].
Obtain relevant dimensionless groups for this problem.

Accepted Answer

In this case, [latex]t= f \left(d, k, ho, C_p, heta_c, heta_s ight)[/latex]. The dimensions of each variable are: [latex]t= T , d= L , k= M L / T heta, C_p= L ^2 / T ^2 heta, heta_c= heta, heta_s= heta[/latex]. There are 7 variables and hence with 4 fundamental dimensions, there will be [latex](7-4)=3[/latex] dimensionless groups. Taking [latex]d, ho, C_p[/latex] and [latex] heta_c[/latex] as the recurring set, then:

[latex]d \equiv L , \quad L =d[/latex],

[latex] ho \equiv M / L ^3, \quad M = ho L ^3= ho d^3[/latex]

[latex] heta_c \equiv heta, \quad heta= heta_c[/latex]

[latex]C_p \equiv L ^2 / T ^2 heta \quad C_p=d^2 / T ^2 heta_c[/latex] and [latex]T ^2=d^2 / C_p heta_c ext { or: } T =d / C_p^{0.5} heta_c^{0.5}[/latex]

Thus: dimensionless group 1: [latex]t / T =t C_p^{0.5} heta_c^{0.5} / d[/latex]

dimensionless group 2: [latex]k T heta / M L =k\left(d / C_p^{0.5} heta_c^{0.5} ight) heta_c /\left( ho d^3 ight) d=k heta_c^{0.5} / C_p^{0.5} ho d^3[/latex]
dimensionless group 3: [latex] heta_s / heta= heta_s / heta_c[/latex]

Question 1.P.18: A sphere, initially at a constant temperature, is immersed i...

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