Question

Hot water at \(90^{\circ} \mathrm{C}\) enters a \(15-\mathrm{m}\) section of a cast iron pipe \((k=52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) whose inner and outer diameters are 4 and \(4.6 \mathrm{~cm}\), respectively, at an average velocity of \(1.2 \mathrm{~m} / \mathrm{s}\). The outer surface of the pipe, whose emissivity is \(0.7\), is exposed to the cold air at \(10^{\circ} \mathrm{C}\) in a basement, with a convection heat transfer coefficient of \(12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Taking the walls of the basement to be at \(10^{\circ} \mathrm{C}\) also, determine \((a)\) the rate of heat loss from the water and \((b)\) the temperature at which the water leaves the basement.

Short answer

Question: Calculate the total heat loss rate from the water and the temperature of the water leaving the basement given the following parameters: - Inner diameter of the pipe: 4 cm - Outer diameter of the pipe: 4.6 cm - Length of the pipe: 15 m - Thermal conductivity of the cast-iron pipe: 52 W/m·K - Hot water enters the pipe at 90°C - Air temperature surrounding the pipe: 10°C - Convection heat transfer coefficient: 12 W/m²·K - Mass flow rate of the water: 0.2 kg/s - Specific heat capacity of water: 4186 J/kg·K Answer: Using the provided data, follow the step-by-step solution to calculate the total heat loss rate from the water, \(q_{loss}\), and the temperature of the water leaving the basement.

Step-by-step solution

Calculate the inner and outer surface areas of the pipe

In order to find the heat loss, we will first find the inner and outer surface areas of the pipe. These can be calculated using the following formulas: \(A_{in}=2\pi r_{in} L\), and \(A_{out}=2\pi r_{out} L\) where \(r_{in}\) and \(r_{out}\) are the radii of the inner and outer diameters, and \(L\) is the length of the pipe. From the given data, we know that the inner diameter is \(4 \mathrm{~cm}\) and the outer diameter is \(4.6 \mathrm{~cm}\). Therefore, the radii are \(r_{in}=2 \mathrm{~cm}\) and \(r_{out}=2.3 \mathrm{~cm}\). The length of the pipe is \(15 \mathrm{~m}\). Putting these values in the formulas, we can find the areas.

Calculate the conduction heat transfer rate through the pipe wall

To find the rate of heat loss through conduction, we'll use the formula for conduction through a cylinder: \(q_{cond}=\frac{2\pi Lk(T_{in}-T_{out})}{ln(r_{out}/r_{in})}\) where \(q_{cond}\) is the conduction heat transfer rate, \(k\) is the pipe's thermal conductivity, \(T_{in}\) and \(T_{out}\) are the temperatures at the inner and outer surfaces of the pipe, and \(ln\) is the natural logarithm. We are given that the hot water enters the pipe at \(90^{\circ} \mathrm{C}\), so \(T_{in} = 90^{\circ} \mathrm{C}\). The surrounding air temperature is \(10^{\circ} \mathrm{C}\), and as the pipe is exposed to this air, \(T_{out} = 10^{\circ} \mathrm{C}\). The thermal conductivity of the cast-iron pipe is \(k = 52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Now, we can plug these values into the formula and calculate the conduction heat transfer rate, \(q_{cond}\).

Calculate the convection heat transfer rate at the outer surface of the pipe

Now, we'll find the rate of heat loss through convection from the outer surface of the pipe to the cold air. For that, we'll use the formula for convection heat transfer: \(q_{conv} = hA_{out}(T_{out}-T_{\infty})\) where \(q_{conv}\) is the convection heat transfer rate, \(h\) is the convection heat transfer coefficient, \(T_{\infty}\) is the air temperature, and \(A_{out}\) is the outer surface area of the pipe. From the given data, we know that the air temperature is \(10^{\circ} \mathrm{C}\), and the convection heat transfer coefficient is \(12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). We have already calculated \(A_{out}\) in Step 1. Plugging these values into the formula, we can find the convection heat transfer rate, \(q_{conv}\).

Calculate the total heat loss rate from the water and the temperature of the water leaving the basement

The total heat loss rate from the water can be found by equating the conduction and convection heat transfer rates: \(q_{loss} = q_{cond} = q_{conv}\) We already calculated \(q_{cond}\) and \(q_{conv}\) in the previous steps, so we can now find the total heat loss rate from the water, \(q_{loss}\). To find the temperature of the water leaving the basement, we'll apply the concept of conservation of energy: the heat loss from the water should equal the heat gained by the surrounding air through convection. Therefore, using the formula for the convection heat transfer rate, we can find the final temperature of the water: \(q_{conv} = m \cdot c \cdot (T_{out}-T_{in})\) where \(m\) is the mass flow rate of the water, and \(c\) is the specific heat capacity of water. We can rearrange this formula to find the temperature of the water leaving the basement, \(T_{out}\), given all other parameters and the previously calculated \(q_{conv}\).

Key concepts

Conduction

Conduction is a method of heat transfer that occurs in solids when there is a temperature difference. Essentially, it involves the transfer of heat through a material without any movement of the substance itself. This process happens as heat causes particles in a solid to vibrate more rapidly, passing kinetic energy to neighboring particles.

For our cast iron pipe, conduction occurs when the hot water inside transfers heat through the pipe's walls to the cooler basement air. We calculate conduction using the formula: \[ q_{cond} = \frac{2\pi Lk(T_{in}-T_{out})}{\ln(r_{out}/r_{in})} \] Here, \(q_{cond}\) is the conduction heat transfer rate, \(L\) is the length of the pipe, \(k\) is the thermal conductivity of the material, and \(T_{in}\) and \(T_{out}\) are the internal and external temperatures, respectively. Understanding this concept is essential for determining how efficiently heat travels through materials.

Convection

Convection involves the transfer of heat through fluids (liquids or gases) when there is a movement of the fluid itself. It occurs when warmer parts of a fluid rise and cooler parts sink, creating a flow that aids in heat transfer.

In the context of the pipe, convection occurs between the outer surface of the pipe and the surrounding cold basement air. The heat from the pipe warms the air in contact with it, causing the air to rise and allowing cooler air to take its place, thus creating a cycle. The rate at which this occurs is given by the formula: \[ q_{conv} = hA_{out}(T_{out}-T_{\infty}) \] Here, \(q_{conv}\) represents the convection heat transfer rate, \(h\) is the convection heat transfer coefficient, \(A_{out}\) is the outer surface area, and \(T_{\infty}\) is the temperature of the air far from the pipe.

• This process efficiently transfers heat from the pipe to the surrounding air.
• The convection coefficient acts as a measure of the effectiveness of this heat transfer.

ThermalConductivity

Thermal conductivity is a property of a material that indicates its ability to conduct heat. Different materials transfer heat at different rates, and thermal conductivity quantifies this difference. For the pipe problem, the cast iron has a thermal conductivity \(k = 52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). This value tells us how readily heat is conducted through the pipe's walls.

A higher thermal conductivity means the material can transfer heat more efficiently. This comes into play in our calculation of conduction heat transfer, highlighting that materials with high thermal conductivity, such as metals, are excellent conductors of heat.

• Thermal conductivity is crucial when choosing materials for any application involving heat transfer.
• The thermal conductivity value is used in the conduction formula to determine heat transfer efficiency.

Understanding the concept helps in designing systems like heating, cooling, and insulation.

SpecificHeatCapacity

Specific heat capacity is the amount of heat required to change a unit mass of a substance by one degree in temperature. It's a measure of a material's ability to absorb heat energy. For water, this value is around \( 4.186 \text{ J/g°C} \), making it very effective at storing and transferring heat.

In the pipe scenario, we use specific heat capacity when determining how much heat the water loses as it flows through. By knowing the mass flow rate of the water and its specific heat capacity, we calculate how its temperature changes due to heat loss: \[ q_{conv} = m \cdot c \cdot (T_{out}-T_{in}) \] Here, \(m\) is the mass flow rate, \(c\) is the specific heat capacity, and \(T_{out} - T_{in}\) is the temperature change.

• Understanding this concept is essential for thermal systems where heat absorption and retention are critical.
• It helps predict how the system's temperature will change under different conditions.

Thus, specific heat capacity is a vital parameter in managing energy efficiency in engineering applications.