An uninsulated, thin-walled pipe of 100-mm diameter is used to transport water to equipment that

An uninsulated, thin-walled pipe of 100-mm diameter is used...

An uninsulated, thin-walled pipe of $100-\mathrm{mm}$ diameter is used to transport water to equipment that operates outdoors and uses the water as a coolant. During particularly harsh winter conditions, the pipe wall achieves a temperature of $-15^{\circ} \mathrm{C}$ and a cylindrical layer of ice forms on the inner surface of the wall. If the mean water temperature is $3^{\circ} \mathrm{C}$ and a convection coefficient of $2000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ is maintained at the inner surface of the ice, which is at $0^{\circ} \mathrm{C}$, what is the thickness of the ice layer?

Key Concepts

Newton’s Law of Cooling (Convection)

This law describes the convective heat transfer between a surface and a fluid, characterized by a convection heat transfer coefficient. In the problem, convection is responsible for transferring heat from the inner surface of the ice to the water, and it sets the boundary condition at the ice-water interface.

Thermal Resistance

Thermal resistance is a measure of a material's opposition to heat flow and is analogous to electrical resistance. In a layered system like the pipe with an ice formation, it is used to combine the effects of conduction through the ice and convection from the water, allowing the overall heat transfer to be quantified and the ice thickness to be determined.

Thermal Conduction in Solids

This concept involves the transfer of heat through a material by molecular diffusion, governed by Fourier’s law. In the context of the problem, it explains how heat moves through the ice layer, which acts as an insulating barrier, and requires using the appropriate formula for steady?state conduction to calculate the thermal resistance of the layer.

Radial Heat Transfer in Cylindrical Coordinates

When dealing with pipes, the geometry is cylindrical, and the conduction of heat must be evaluated using formulas specific to cylinders. This concept is essential for determining how the ice layer’s thickness affects the temperature gradient from the cold exterior to the warm water inside the pipe.

Phase Change and Equilibrium Temperature

When water freezes to form ice, the phase change occurs at the equilibrium or melting temperature (0°C for water at standard pressure). The constant temperature at the ice-water interface is a critical boundary condition in this problem, as it defines the temperature at which the phase change occurs and influences the heat transfer analysis.

Transcript

00:01 In this problem the objective is to find the thickness of the ice layer and the diameter of the pipe denoted by d not will be equal to it is here hundred and then it will be milliliter and this is equal to zero point one now here it is meter so the mean water temperature denoted by t infinity will be equal to it is here three degrees and this will be here equal to it is here 3 plus 273 and this will be here equals to 276 kelvin.

00:39 So let us talk about the convection heat transfer coefficient denoted by h and it is equals to 2000 and then it is watt.

00:49 Now it will be per meter square kelvin.

00:53 So it's time to talk about the inner surface temperature.

00:57 So it will be t s and here it is comma i and this value will be equals to 0 degree celsius.

01:06 So this is equal to 0 plus 273 and here it will be kelvin.

01:13 So this value will be equals to 273 kelvin.

01:18 So it's time to write the outer surface temperature which can be written as t s and here it will be o.

01:25 So this is equal to negative 15 degrees celsius.

01:30 So this can be here converted into kelvin and it will be negative 15 plus 273.

01:37 So this value will be equals to 258 kelvin.

01:43 So it's time to write the value of thermal conductivity of ice.

01:48 And it is considered to be equals to here 1 .94 and we have obtained this value from the table a .3 and here it can be written as such.

02:01 So let us draw the circuit diagram for thermal resistance and it can be here denoted as here we will be having the resistance and this is the resistance of convection and then here we will be having the resistance of conduction and here the temperature will be t s and here it is o that is the outer surface temperature and at this point we will be having the inner surface temperature and here we will be having the temperature which is basically called as the mean water temperature.

02:39 So let us write here that this is our r dash convection and then here it is r dash conduction.

02:49 So let's see how we can solve this very problem further and we can assume that let us suppose the inner diameter of the ice layer is denoted by d .i...

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