Heat Transfer Coefficients in Nuclear Reactors

Heat Transfer Coefficients in Nuclear Reactors

• Heat removal by coolant can occur through conduction and convection.
• Conduction dominates in laminar flow where fluid moves parallel to the channel with no radial movement.
• Convection dominates in turbulent flow with significant velocity fluctuations in the radial direction, carrying heat from the wall to the bulk of the liquid.
• Nuclear power plants utilize forced convection, creating a turbulent flow with a more uniform bulk temperature and causing a rapid temperature drop near the fuel.

Reynolds Number

• This dimensionless parameter characterizes fluid flow, relating fluid velocity, density, and viscosity to channel size.
• The Reynolds number is calculated with the formula: Re = (De * u * p) / u, where De is the equivalent diameter of the channel, u is the velocity, p is the density, and u is the viscosity.
• Turbulent flow, indicated by high Re values, leads to greater heat transfer to the coolant.
• The following conditions are used to classify flow:
  • Laminar flow: Re < 2000
  • Mixed flow: 2000 < Re < 10000
  • Turbulent flow: Re > 10000

Heat Transfer Coefficients in Nuclear Reactors

• Heat transfer coefficients are determined experimentally and expressed as experimental correlations.
• The Nusselt number (Nu) is used to describe heat transfer effectiveness, with Nu = (h*De)/k, where h is the heat transfer coefficient, De is the equivalent diameter, and k is the thermal conductivity.
• The Prandtl number (Pr) is used to describe the relative importance of momentum and thermal diffusion in fluid flow, with Pr = (cp*u)/k, where cp is the specific heat, u is the viscosity, and k is the thermal conductivity.
• For turbulent flow in a long straight channel, the Nusselt number can be calculated using the Dittus-Boelter equation: Nu = C * Re^m * Pr^n, where C, m, and n are constants based on the specific fluid and flow conditions.
• The Dittus-Boelter equation has limitations:
  • Only valid for reference temperatures.
  • Not applicable when conduction becomes significant.
  • Large errors can occur if the channel deviates significantly from a circular shape.
• Liquid metal coolants, such as sodium, have high thermal conductivities, resulting in less steep temperature gradients in the coolant.
• A specific correlation is available for liquid metal coolants through a hexagonal lattice of rods: Nu = 0.66 + 3.126*(s/d) + 1.184*(s/d)^2 + 0.0155*(Pe)^0.86, where s/d is the ratio of lattice pitch to rod diameter, and Pe is the Peclet number.
• The Peclet number, Pe, is calculated as Pe = Re * Pr.

Newton's Law of Cooling

• Newton's law of cooling describes the heat transfer from a heated solid to a moving fluid, with the formula: q" = h * (Tc - Tb), where q" is the heat flux, h is the heat transfer coefficient, Tc is the surface temperature, and Tb is the bulk temperature of the fluid.
• Heat transfer coefficients typically vary based on the fluid, flow type, coolant temperature, etc.

Total Thermal Resistance

• The total thermal resistance in a system can be calculated as the sum of the resistances associated with each component.
• For a plate-type fuel element, the total resistance is R = (a/2kA) + (b/kcA) + (1/h*A), where k is the thermal conductivity, a and b are thicknesses, and A is the area.
• For a clad cylindrical fuel element, the total resistance is R = (1/4pikfH) + (ln(1+b/a)/2pikcH) + (1/h*A), where kf is the fuel conductivity, kc is the cladding conductivity, a and b are radii, and H is the height.
• For a space-dependent heat source, the heat transfer rate can be calculated for specific points along the channel using the formula: q(z) = (T(z)-Tb(z))/R, where T(z) and Tb(z) are the temperatures at position z, and R is the total resistance.

Temperature along the Coolant Channel

• The coolant channel is a volume associated with a single fuel rod where heat generated within the fuel transfers to the channel.
• As heat is added, the coolant temperature increases, causing a temperature rise along the coolant channel.
• The temperature change in a specific section of coolant can be calculated as dTb = (q" * Af * dz)/(w * cp), where q" is the heat flux, Af is the fuel surface area, dz is the length of the coolant section, w is the coolant flow rate, and cp is the specific heat of the coolant.
• For a fuel rod at the center of the channel, the heat flux is calculated as q'" = qmax * cos(pi * z / H), where qmax is the maximum heat generation rate, z is the position along the channel, and H is the channel length.
• The maximum bulk temperature of the coolant in the channel is calculated as: Tb = Tb0 + (qmax * Af / w * cp * pi) * [1 + sin(pi * z / H)] / [cos(pi * z / H)], where Tb0 is the initial coolant temperature, qmax is the maximum heat generation rate, Af is the fuel surface area, w is the coolant flow rate, cp is the specific heat of the coolant, z is the position along the channel, and H is the channel length.
• The temperature difference between the mean coolant temperature (Tm) and bulk temperature (Tb) can be calculated as: Tm - Tb = R * qmax * cos(pi * z / H) / H, where R is the total thermal resistance, qmax is the maximum heat generation rate, z is the position along the channel, and H is the channel length.
• The maximum coolant temperature (Tb) along the channel can be calculated as: Tb = Tb0 + (qmax / pi * w * cp) * [1 + sin(pi * z / H)] / [cos(pi * z / H)], where Tb0 is the initial coolant temperature, qmax is the maximum heat generation rate, w is the coolant flow rate, cp is the specific heat of the coolant, z is the position along the channel, and H is the channel length.
• The temperature difference between the surface temperature (Tc) and bulk temperature (Tb) is calculated as: Tc - Tb = Rh * qmax * cos(pi * z / H) / H, where Rh is the thermal resistance of the coolant, qmax is the maximum heat generation rate, z is the position along the channel, and H is the channel length.