Question

Steam exiting the turbine of a steam power plant at \(100^{\circ} \mathrm{F}\) is to be condensed in a large condenser by cooling water flowing through copper pipes \(\left(k=223 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)\) of inner diameter \(0.4\) in and outer diameter \(0.6\) in at anerage temperature of \(70^{\circ} \mathrm{F}\). The heat of vaporization of water at \(100^{\circ} \mathrm{F}\) is \(1037 \mathrm{Btu} / \mathrm{lbm}\). The heat transfer coefficients are \(1500 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) on the steam side and \(35 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) on the water side. Determine the length of the tube required to condense steam at a rate of \(120 \mathrm{lbm} / \mathrm{h}\). Answer: \(1150 \mathrm{ft}\)

Short answer

Based on the analysis and step-by-step solution above, the length of the copper tube required to condense steam at a rate of 120 lbm/h with given heat transfer coefficients and temperature differences is approximately 1150 feet.

Step-by-step solution

Calculate the heat transfer rate

In order to find the length of the copper tube required to condense steam at a specific rate, we first need to calculate the heat transfer rate. We can find the heat transfer rate (Q) by multiplying the mass flow rate of steam (m) and its heat of vaporization (h). So, Q = m * h where: Q = heat transfer rate (Btu/h) m = mass flow rate of the steam (120 lbm/h) h = heat of vaporization of water at 100°F (1037 Btu/lbm) Q = 120 lbm/h × 1037 Btu/lbm = 124440 Btu/h

Calculate individual thermal resistances

Now let's analyze the three thermal resistances which are involved in the process: Resistance on the steam side (R1), resistance of the copper tube (R2), and resistance on the water side (R3). The formulas for calculating these are given by: R1 = 1/(h1 * A1) R2 = ln(r2/r1)/(2 * π * k * L) R3 = 1/(h2 * A2) where: R1, R2, R3 = thermal resistances (h·ft·°F/Btu) h1 = heat transfer coefficient on the steam side (1500 Btu / h·ft²·°F) A1 = area of steam side (π * D1 * L) h2 = heat transfer coefficient on the water side (35 Btu / h·ft²·°F) A2 = area of water side (π * D2 * L) r1 = inner radius of the copper tube (0.2 in) r2 = outer radius of the copper tube (0.3 in) k = thermal conductivity of copper (223 Btu / h·ft·°F) D1 = inner diameter of the copper tube (0.4 in) D2 = outer diameter of the copper tube (0.6 in) L = length of the copper tube (ft)

Find the overall heat transfer coefficient

After calculating the individual thermal resistances, we can find the overall heat transfer coefficient (U) by taking the reciprocal of the total thermal resistance (R_total). R_total = R1 + R2 + R3 U = 1/R_total

Calculate the required length of the copper tube

Since we've already calculated the heat transfer rate (Q) and the overall heat transfer coefficient (U), we can now find the required length of the copper tube (L) by using the formula: Q = U * A * ΔT where: A = heat transfer area = π * D1 * L (ft²) ΔT = driving temperature difference between steam and water (100°F - 70°F) Rearranging the formula to solve for L: L = Q / (U * π * D1 * ΔT) When plugging in the values, make sure to convert the diameters into feet (1 in = 1/12 ft). Simplify and solve for L: L = 124440 Btu/h / (U * π * 0.4/12 ft * 30°F) Upon substituting the value of U obtained in Step 3, we find that: L ≈ 1150 ft Therefore, the length of the copper tube required to condense steam at a rate of 120 lbm/h is approximately 1150 feet.

Key concepts

Thermal Conductivity

Thermal conductivity is a measure of a material's ability to conduct heat. Highly conductive materials can transfer heat more efficiently, making them very important in engineering applications, such as heat exchangers.
A common example is copper, with a thermal conductivity of about 223 Btu/h·ft·°F. This makes copper an excellent choice for tubes and pipes in systems where heat needs to be conducted away quickly.

• The unit of measurement for thermal conductivity in the Imperial system is Btu/h·ft·°F, which describes how much heat (in Btu) can pass through a material of a given thickness per hour, based on the temperature gradient across it.
• Factors affecting thermal conductivity include the type of material and its temperature. Pure copper will have different thermal conductivity compared to brass or aluminum.

In our steam power plant example, copper pipes are used because of their high thermal conductivity, which facilitates efficient heat transfer from the steam to the cooling water, enabling effective condensation.

Heat Exchangers

Heat exchangers are devices that transfer heat between two or more fluids without them mixing. They are essential in many engineering systems, such as heating and cooling buildings, automobiles, and power plants.
The primary purpose of a heat exchanger is to efficiently transfer heat from one medium to another, often to improve energy savings, efficiency, or process stability.

• In our example, the copper tubes act as a heat exchanger where steam inside the tubes releases its heat to surrounding cooling water, causing the steam to condense into water.
• Heat exchangers can come in various configurations, including shell and tube, plate, and finned tube designs. Each design caters to specific system requirements and constraints.

The effectiveness of a heat exchanger is determined by parameters such as the surface area available for heat exchange, the flow rates of the fluids involved, and the temperature difference between them. Copper's high thermal conductivity means the heat exchanger in this application can be compact while still being highly effective.

Condensation Process

Condensation is the physical process of turning vapor into liquid. This typically occurs when a vapor is cooled to a temperature below its dew point, causing it to release heat during this phase change. In the steam power plant scenario, the steam exits the turbine and enters a condenser where it loses heat to the cooling water surrounding the copper tubes.
During condensation:

• Energy is released as the latent heat of vaporization when the steam transitions to liquid. In our problem, this value is given as 1037 Btu/lbm.
• The process is dictated by temperature and pressure conditions. As the steam cools down below its saturation temperature, it begins to change phase.
• The rate of condensation is influenced by the efficiency of the heat exchanger and the driving temperature difference between the steam and the cooling water.

Effective condensation is crucial for power plants to cycle efficiently, as it allows steam to be converted back into water that can be reheated in the boiler, maintaining a continuous power generation cycle.