Transient pressure and compressibility effects are also not included.

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)

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 Q_n is based on fully developed laminar flow.

Bernoulli's equation is:

(13–175)

where:

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:

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 = v_a, 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 (P_I - P_J) 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 B_c P_L.

Two further points: