13.116. FLUID116 - Coupled Thermal-Fluid Pipe

Matrix or VectorGeometryShape Functions Integration Points
Thermal Conductivity MatrixBetween nodes I and J Equation 11–13 None
Convection between nodes I and K and between nodes J and L (optional)NoneNone
Pressure Conductivity Matrix Between nodes I and J Equation 11–12 None
Specific Heat Matrix and Heat Generation Load Vector Equation 11–13 None

13.116.1. Assumptions and Restrictions

Transient pressure and compressibility effects are also not included.

13.116.2. Combined Equations

The thermal and pressure aspects of the problem have been combined into one element having two different types of working variables: temperatures and pressures. The equilibrium equations for one element have the form of:

(13–167)

where:

[Ct] = specific heat matrix for one channel
{T} = nodal temperature vector
{P} = nodal pressure vector
[Kt] = thermal conductivity matrix for one channel (includes effects of convection and mass transport)
[Kp] = pressure conductivity matrix for one channel
{Q} = nodal heat flow vector (input as HEAT on F command)
{w} = nodal fluid flow vector (input as FLOW on F command)
{Qg} = internal heat generation vector for one channel
{H} = gravity and pumping effects vector for one channel
Nc = number of parallel flow channels (input as Nc on R command)

13.116.3. Thermal Matrix Definitions

Specific Heat Matrix

The specific heat matrix is a diagonal matrix with each term being the sum of the corresponding row of a consistent specific heat matrix:

(13–168)

where:

ρ = mass density (input as DENS on MP command
P = pressure (average of first two nodes)
Tabs = T + TOFFST = absolute temperature
T = temperature (average of first two nodes)
TOFFST = offset temperature (input on TOFFST command)
Cp = specific heat (input as C on MP command)
A = flow cross-sectional area (input as A on R command)
Lo = length of member (distance between nodes I and J)
Rgas = gas constant (input as Rgas on R command)

Thermal Conductivity Matrix

The thermal conductivity matrix is given by:

(13–169)

where:

Ks = thermal conductivity (input as KXX on MP command)
B2 = h AI
h = film coefficient (defined below)
B3 = h AJ
D = hydraulic diameter (input as D on R command)
w = mass fluid flow rate in the element

w may be determined by the program or may be input by the user:

(13–170)

The above definitions of B4 and B5, as used by Equation 13–169, cause the energy change due to mass transport to be lumped at the outlet node.

The film coefficient h is defined as:

(13–171)

Nu, the Nusselt number, is defined for KEYOPT(4) = 1 as:

(13–172)

where:

N1 to N4 = input constants (input on R commands)
μ = viscosity (input as VISC on MP command)

A common usage of Equation 13–172 is the Dittus-Boelter correlation for fully developed turbulent flow in smooth tubes (Holman([55])):

(13–173)

where:

Heat Generation Load Vector

The internal heat generation load vector is due to both average heating effects and viscous damping:

(13–174)

where:

VDF = viscous damping multiplier (input on RMORE command)
Cver = units conversion factor (input on RMORE command)
v = average velocity

The expression for the viscous damping part of Qn is based on fully developed laminar flow.

13.116.4. Fluid Equations

Bernoulli's equation is:

(13–175)

where:

Z = coordinate in the negative acceleration direction
P = pressure
γ = ρg
g = acceleration of gravity
v = velocity
PPMP = pump pressure (input as Pp on R command)
CL = loss coefficient

The loss coefficient is defined as:

(13–176)

where:

a = additional length to account for extra flow losses (input as La on R command)
k = loss coefficient for typical fittings (input as K on R command)
f = Moody friction factor, defined below:

For the first iteration of the first load step,

(13–177)

where:

fm = input as MU on MP command

For all subsequent iterations

(13–178)

The smooth pipe empirical correlation is:

(13–179)

Bernoulli's Equation 13–175 may be simplified for this element, since the cross-sectional area of the pipe does not change. Therefore, continuity requires all velocities not to vary along the length. Hence v1 = v2 = va, so that Bernoulli's Equation 13–175 reduces to:

(13–180)

Writing Equation 13–180 in terms of mass flow rate (w = ρAv), and rearranging terms to match the second half of Equation 13–167,

(13–181)

Since the pressure drop (PI - PJ) is not linearly related to the flow (w), a nonlinear solution will be required. As the w term may not be squared in the solution, the square root of all terms is taken in a heuristic way:

(13–182)

Defining:

(13–183)

and

(13–184)

Equation 13–182 reduces to:

(13–185)

Hence, the pressure conductivity matrix is based on the term and the pressure (gravity and pumping) load vector is based on the term Bc PL.

Two further points:

  1. Bc is generalized as:

    (13–186)

  1. (-ZI + ZJ)g is generalized as:

    (13–187)

    where:

    {Δx} = vector from node I to node J
    {at} = translational acceleration vector which includes effects of angular velocities (see Acceleration Effect)


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