Question

Exposure to high concentration of gaseous ammonia can cause lung damage. To prevent gaseous ammonia from leaking out, ammonia is transported in its liquid state through a pipe \(\left(k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i, \text { pipe }}=2.5 \mathrm{~cm}\right.\), \(D_{o, \text { pipe }}=4 \mathrm{~cm}\), and \(\left.L=10 \mathrm{~m}\right)\). Since liquid ammonia has a normal boiling point of \(-33.3^{\circ} \mathrm{C}\), the pipe needs to be properly insulated to prevent the surrounding heat from causing the ammonia to boil. The pipe is situated in a laboratory, where air at \(20^{\circ} \mathrm{C}\) is blowing across it with a velocity of \(7 \mathrm{~m} / \mathrm{s}\). The convection heat transfer coefficient of the liquid ammonia is \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Calculate the minimum insulation thickness for the pipe using a material with \(k=0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) to keep the liquid ammonia flowing at an average temperature of \(-35^{\circ} \mathrm{C}\), while maintaining the insulated pipe outer surface temperature at \(10^{\circ} \mathrm{C}\).

Short answer

Answer: The minimum insulation thickness required is approximately 6.45 mm.

Step-by-step solution

Calculate the heat transfer through the insulated pipe

First, we need to find the heat transfer through the insulation material. This can be calculated using the formula: $$q = \frac{2 \pi k_i L (T_{ave} - T_{s, out})}{\ln \left(\frac{D_{o, ins}}{D_{o, pipe}}\right)}$$ Where: \(q\): Heat transfer through insulation [W] \(k_i\): Insulation material thermal conductivity [W/m·K] = 0.75 W/m·K \(L\): Pipe length [m] = 10 m \(T_{ave}\): Average temperature of liquid ammonia [°C] = -35°C \(T_{s, out}\): Outer surface temperature of insulated pipe [°C] = 10°C \(D_{o, ins}\): Outer diameter of insulation [m] \(D_{o, pipe}\): Outer diameter of the pipe [m] = 0.04 m In this step, we still need to find \(D_{o, ins}\), which is the outer diameter of insulation and depends on the insulation thickness.

Calculate the heat transfer through the pipe

Now we need to find the heat transfer through the pipe. Using the convection heat transfer formula: $$q = h A (T_{s, out} - T_{amb})$$ Where: \(h\): Convection heat transfer coefficient [W/m²·K] = 100 W/m²·K \(A\): Outer surface area of the pipe [m²] = \(2 \pi D_{o, pipe} L\) \(T_{amb}\): Ambient temperature [°C] = 20°C Since the insulation is effective, the heat transfer through the pipe should be equal to the heat transfer through the insulation. So, we can equate the heat transfer formulas for pipe and insulation.

Equate heat transfer formulas and solve for insulation thickness

The heat transfer through the insulation and pipe are equal, so: $$\frac{2 \pi k_i L (T_{ave} - T_{s, out})}{\ln \left(\frac{D_{o, ins}}{D_{o, pipe}}\right)} = h 2 \pi D_{o, pipe} L (T_{s, out} - T_{amb})$$ Now, we can solve for insulation thickness: $$\frac{k_i (T_{ave} - T_{s, out})}{\ln \left(\frac{D_{o, ins}}{D_{o, pipe}}\right)} = h D_{o, pipe} (T_{s, out} - T_{amb})$$ $$D_{o, ins} = D_{o, pipe} \cdot \exp \left(\frac{k_i (T_{ave} - T_{s, out})}{h D_{o, pipe} (T_{s, out} - T_{amb})}\right)$$ Plugging in the given values: $$D_{o, ins} = 0.04 m \cdot \exp \left( \frac{0.75 ( -35 - 10 )}{100 \cdot 0.04 ( 10 - 20)}\right)$$ $$D_{o, ins} \approx 0.0529 \; m$$ Finally, we calculate the minimum insulation thickness: $$Insulation \; Thickness = \frac{D_{o, ins} - D_{o, pipe}}{2}$$ $$Insulation \; Thickness \approx \frac{0.0529 - 0.04}{2} \approx 0.00645 \; m \; (6.45 \; mm)$$ Therefore, the minimum insulation thickness needed to keep the liquid ammonia flowing at an average temperature of \(-35^{\circ} \mathrm{C}\) while maintaining the insulated pipe outer surface temperature at \(10^{\circ} \mathrm{C}\) is approximately 6.45 mm.

Key concepts

Heat Transfer in Pipes

Understanding the dynamics of heat transfer in pipes is crucial in various engineering applications, such as in the transportation of chemicals. Heat transfer is essentially the movement of heat from a region of high temperature to one of low temperature. In the context of pipes, heat transfer can occur through three main mechanisms: conduction, convection, and radiation. For liquids in pipes, such as in the transportation of liquid ammonia, the primary concern is usually the conductive and convective heat transfers.
Conductive heat transfer in pipes involves the heat moving through the pipe material itself and any insulation layer that may be present. This requires understanding the thermal properties of the substances involved. For the convective heat transfer, it's usually about managing the heat exchange between the pipe surface and the surrounding environment, which is affected by factors like air temperature and velocity.
Optimizing the insulation thickness is one example of controlling heat transfer. It ensures that the temperature of the transported fluid remains within desired limits, thereby preventing issues such as boiling for sensitive chemicals like ammonia. The insulation acts as a barrier, slowing down the heat flow from the warm exterior to the cold fluid inside the pipe.

Thermal Conductivity

The term thermal conductivity, denoted by 'k', is a material-specific property that measures how well a material conducts heat. It is defined as the amount of heat that passes through a material of given thickness and surface area, due to a temperature difference across the material. It is usually expressed in units of watts per meter-kelvin (W/m·K).
Materials with high thermal conductivity, such as metals, are good heat conductors, making them unsuitable for insulation purposes. Conversely, materials with low thermal conductivity, such as foam or certain plastics, are effective at insulating because they impede heat flow. The choice of insulation material in the exercise, with a thermal conductivity of 0.75 W/m·K, reflects the need for a substance that can limit the amount of heat transferred from the warmer exterior to the cooler liquid ammonia within the pipe.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient, commonly symbolized by 'h', quantifies the convective heat transfer occurring between a surface and a fluid moving over it. It encompasses both the properties of the fluid and the nature of the flow—whether it is turbulent or laminar, among other factors. The value of 'h' is given in watts per square meter-kelvin (W/m²·K).
This coefficient is a determining factor for how effectively heat is exchanged between the pipe and its environment. A higher 'h' value means a greater rate of heat exchange, which is something you'd want to minimize in the case of transporting liquid ammonia. Since the laboratory surroundings in the exercise have a particular 'h' value, the calculations for insulation thickness must ensure that this exchange is sufficiently mitigated to maintain the ammonia below its boiling point.

Boiling Point of Ammonia

The boiling point of ammonia is a critical thermal characteristic, especially when it needs to be transported in liquid form. Ammonia boils at -33.3 degrees Celsius at atmospheric pressure, which is a relatively low temperature compared to water. This means that if the temperature of liquid ammonia in a pipe rises above its boiling point, the ammonia would vaporize and could potentially cause safety hazards like pipe ruptures or leaks, as well as health risks due to ammonia's toxicity when inhaled.
Therefore, in scenarios like the presented exercise, where ammonia must remain cool, the insulation's effectiveness is essential. Not only must the insulation prevent external heat from warming the contents of the pipe, but it must also ensure that such a low boiling point is never reached, maintaining the ammonia in a safe, liquid state throughout its transport.