Shell-and-Tube Heat Exchanger
A shell-and-tube heat exchanger is a commonly used heat transfer device that effectively cools or heats fluids by allowing them to flow through a series of tubes enclosed within a shell. The fluid flowing inside the tubes is generally called the tube-side fluid, and the fluid outside the tubes but inside the shell is referred to as the shell-side fluid.
The design of a shell-and-tube heat exchanger involves several components such as baffles, which direct the shell-side fluid flow and improve heat transfer, and tube bundles, which contain multiple tubes to maximize the surface area for heat exchange. In our example, we consider an exchanger cooling a process stream using water as the coolant on the shell side. One critical aspect of design is determining the most efficient tube length, which ensures that the desired temperature change is achieved without excessive construction costs or pressure losses.
Heat Transfer Calculations
Heat transfer calculations are vital in determining the overall performance of a heat exchanger. These calculations take into account the amount of heat energy that needs to be transferred between the two fluids to achieve the required temperature changes. The basic formula used in these calculations is the heat transfer equation,
\(Q = \dot{m}_p c_{p, p} \Delta T_p\),
where \(Q\) represents the heat transfer rate, \(\dot{m}_p\) is the mass flow rate of the process stream, \(c_{p, p}\) is the specific heat capacity of the process stream, and \(\Delta T_p\) is the temperature difference of the process stream. The result gives us a quantitative measure of the cooling capacity required by the exchanger and is the starting point for further design decisions.
Nusselt Number
The Nusselt number (Nu) is a dimensionless number representing the ratio of convective to conductive heat transfer at a boundary in a fluid medium. In essence, it provides a measure of the relative effectiveness of these two mechanisms in transferring heat.
Higher Nusselt numbers indicate greater convective heat transfer relative to conduction, which typically happens with turbulent flow or when certain geometries enhance fluid mixing. When designing heat exchangers, engineers use the Nusselt number to compute the convective heat transfer coefficient. The formula for Nu can vary depending on the flow conditions and physical properties of the fluid, but a common way to calculate it for a tube flow, as shown in our example, is using the Sieder-Tate correlation. This correlation factors in the Reynolds and Prandtl numbers of the fluid, which reflect its flow dynamics and thermal properties, respectively.
Reynolds Number
The Reynolds number (Re) is another dimensionless number that is used to predict the flow regime of the fluid within the tubes of the heat exchanger. It fundamentally describes whether the flow is laminar or turbulent.
The Reynolds number is calculated using the formula:
\(Re = \frac{\rho \cdot V \cdot D_i}{\mu}\),
where \(\rho\) is the fluid density, \(V\) is the velocity of the fluid inside the tube, \(D_i\) is the inside diameter of the tube, and \(\mu\) is the dynamic viscosity of the fluid. Laminar flow typically occurs at low Reynolds numbers and is characterized by smooth, orderly fluid motion, whereas turbulent flow occurs at high Reynolds numbers and involves chaotic, irregular fluid motion, which enhances heat transfer.
Prandtl Number
The Prandtl number (Pr) is a dimensionless number that relates the momentum diffusivity (kinematic viscosity) to the thermal diffusivity of a fluid. It essentially compares the relative thickness of the velocity boundary layer to the thermal boundary layer at a wall.
It is defined as:
\(Pr = \frac{c_{p} \mu}{k}\),
where \(c_p\) is the specific heat of the fluid at constant pressure, \(\mu\) is the dynamic viscosity, and \(k\) is the thermal conductivity of the fluid. The Prandtl number gives insight into the behavior of the fluid's heat transfer; fluids with high Pr numbers have a relatively low thermal diffusivity resulting in a thicker thermal boundary compared to their velocity boundary.
Heat Transfer Coefficient
The heat transfer coefficient is a critical parameter in calculating the heat exchange in a heat exchanger. It relates to the ease with which heat is transferred between the fluid and the surface of the tubes. In our example, the tube-side heat transfer coefficient is estimated using the Sieder-Tate equation that includes the Nusselt number.
As the equation shows, the heat transfer coefficient (h_t) is a function of the tube's thermal conductivity (k_t), the inside diameter (D_i), and the Nusselt number (Nu). With a higher heat transfer coefficient, more heat can be transferred per unit area at a given temperature difference, which is desirable for efficient heat exchanger operation. When both turbulent flow (high Reynolds number) and good thermal properties (high Prandtl number) are present, larger heat transfer coefficients are achieved, optimizing the design and efficiency of the heat exchanger.
