Question

Consider a \(10-\mathrm{m}\)-long smooth rectangular tube, with \(a=50 \mathrm{~mm}\) and \(b=25 \mathrm{~mm}\), that is maintained at a constant surface temperature. Liquid water enters the tube at \(20^{\circ} \mathrm{C}\) with a mass flow rate of \(0.01 \mathrm{~kg} / \mathrm{s}\). Determine the tube surface temperature necessary to heat the water to the desired outlet temperature of \(80^{\circ} \mathrm{C}\).

Short answer

Answer: The required surface temperature of the tube is 113.5°C.

Step-by-step solution

Calculate the volume flow rate and average velocity of water

First, we need to determine the volume flow rate (Q) and the average velocity (V_avg) of water in the tube. To find Q, we use: Q = (mass flow rate) / (density of water) For water at 20°C, the density is approximately 998 kg/m³. Thus, Q = 0.01 kg/s ÷ 998 kg/m³ = 0.00001 m³/s Now, let's calculate the average velocity (V_avg). The cross-sectional area (A) of the rectangular tube is: A = a * b = (0.05 m)(0.025 m) = 0.00125 m² And the average velocity is: V_avg = Q / A = 0.00001 m³/s ÷ 0.00125 m² = 0.008 m/s

Calculate the Reynolds number

Next, we will calculate the Reynolds number (Re) to characterize the flow regime. The Reynolds number is given by: Re = (density * V_avg * hydraulic diameter) / dynamic viscosity The hydraulic diameter (D_h) for a rectangular tube is: D_h = 2ab/(a+b) = 2(0.05 m)(0.025 m) / (0.05 m + 0.025 m) = 0.0333 m For water at 20°C, the dynamic viscosity is approximately 0.001002 kg/m·s. Therefore, Re = (998 kg/m³)(0.008 m/s)(0.0333 m) / 0.001002 kg/m·s = 265.9 Since Re < 2000, the flow is considered laminar.

Calculate the Nusselt number and the convective heat transfer coefficient

For laminar flow in a rectangular tube, the Nusselt number (Nu) is estimated using the following equation: Nu = 3.66 Next, we calculate the convective heat transfer coefficient (h) using: h = (Nu * thermal conductivity of water) / D_h For water at average temperature of 50°C, the thermal conductivity is approximately 0.617 W/m·K. Thus, h = (3.66)(0.617 W/m·K) / 0.0333 m = 67.6 W/m²·K

Calculate the heat transfer rate

Now, let's find the heat transfer rate (Q_total) required to heat the water from 20°C to 80°C. We use: Q_total = (mass flow rate) * (specific heat of water) * (temperature difference) For water at 20°C, the specific heat is approximately 4182 J/kg·K. So, Q_total = (0.01 kg/s)(4182 J/kg·K)(80°C - 20°C) = 2510 J/s or 2510 W

Determine the tube surface temperature

Finally, we can find the tube surface temperature (T_s) using the following equation for the heat transfer rate: Q_total = h * A_total * (T_s - T_water_avg) Where A_total is the total surface area of the tube, and T_water_avg is the average water temperature, which is (20°C + 80°C) / 2 = 50°C. The total surface area (A_total) of the tube is: A_total = 2(ab + ac) * tube_length = 2(0.05 m * 0.025 m + 0.05 m * 0.025 m) * 10 m = 0.05 m² By plugging the values into the equation, we get: 2510 W = (67.6 W/m²·K)(0.05 m²)(T_s - 50°C) Now we only need to solve for T_s: T_s = (2510 W ÷ (67.6 W/m²·K)(0.05 m²)) + 50°C = 113.5°C The tube surface temperature necessary to heat the water to the desired outlet temperature of 80°C is 113.5°C.

Key concepts

Convective Heat Transfer Coefficient

The convective heat transfer coefficient, often denoted as \(h\), is a measure of a fluid's ability to transfer heat to or from a surface. It plays a crucial role in calculating the rate of heat transfer between the surface and the fluid moving past it. In essence, \(h\) depends on several factors like fluid velocity, viscosity, and surface geometry. This coefficient is crucial when dealing with systems such as heat exchangers, radiators, and other thermal systems.

To calculate \(h\), you need to know the Nusselt number \(\text{Nu}\), a dimensionless number that characterizes convective heat transfer. The relation follows the equation:
\[ h = \frac{\text{Nu} \cdot k}{D_h} \]
where \(k\) is the thermal conductivity of the fluid, and \(D_h\) is the hydraulic diameter. In the context of the original problem, \(h\) was determined to be 67.6 W/m²·K using the calculated Nusselt number for laminar flow conditions in a rectangular tube.

• The higher the value of \(h\), the more efficient the heat transfer process.
• Factors that can increase \(h\) include higher fluid velocity and better fluid mixing.
• This coefficient is vital for designing efficient thermal systems by ensuring sufficient heat transfer rates.

Nusselt Number

The Nusselt number \(\text{Nu}\) is a dimensionless number that quantifies the enhancement of heat transfer through a fluid layer as a result of convection compared to conduction across the same fluid layer. It is essentially the ratio of conduction to convection heat transfer in a fluid. In practical terms, \(\text{Nu}\) allows engineers to estimate the convective heat transfer coefficient indirectly, leading to better-designed thermal systems.
In the exercise, the Nusselt number for the laminar flow in a rectangular tube was estimated using a simplified equation \(\text{Nu} = 3.66\), which is typical for thermally developing laminar flows. This simplified model is possible due to the specific geometry and flow conditions.

• For laminar flow, \(\text{Nu}\) can often be constant, which simplifies calculations.
• In turbulent flows, \(\text{Nu}\) becomes a function of other parameters like Reynolds and Prandtl numbers.
• A higher \(\text{Nu}\) indicates better convective heat transfer performance.

Understanding how \(\text{Nu}\) is influenced by different factors like flow regime and temperature gradients is crucial for engineers to predict and enhance thermal system performance.

Reynolds Number

The Reynolds number \(\text{Re}\) is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is calculated using the equation:
\[ \text{Re} = \frac{\rho V_{avg} D_h}{\mu} \]
where \(\rho\) is the fluid density, \(V_{avg}\) is the average fluid velocity, \(D_h\) is the hydraulic diameter, and \(\mu\) is the dynamic viscosity.
The Reynolds number indicates whether the flow is laminar or turbulent. In the exercise, the flow inside the tube has a Reynolds number of 265.9, which signifies that the flow is laminar, as it is less than 2000. Understanding whether a flow is laminar or turbulent affects how heat transfer calculations are conducted and which formulas are applicable.

• A low \(\text{Re}\) (typically less than 2000) indicates laminar flow, where fluid moves in parallel layers.
• A high \(\text{Re}\) (over 4000) signifies turbulent flow, characterized by chaotic changes in pressure and velocity.
• This classification helps determine appropriate modeling for flow and heat transfer predictions.

Knowing \(\text{Re}\) allows engineers to identify the nature of the flow regime, essential for designing fluid flow and heat exchange systems.