7.6 Forced Fluid Convection

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In most cases, the flux boundary conditions are made up of the terms in Equations (7.23)

and (7.24). Equation (7.23) can also be written as

qk

conv

= h(θ)(θ − θe)nk (7.55)

HEAT TRANSFER 313

T

Tj Ti

Tk

Tl

˙m

ij

˙m

ik

˙m

il

h(T − Ti)

Figure 7.1 Heat fluxes from and toward location i

where θe is the absolute temperature of the surrounding fluid. In some applications, such

as in tubes with internal flow, this temperature is itself also an unknown, depending on

the fluid temperature at the entry of the tube. In such cases, the fluid temperature can be

calculated using a simple network. For applications in which the fluid is meshed with finite

elements, see (Reddy and Gartling 2001).

Consider the part of the tube wall shown in Figure 7.1. The relative material temperature

T interacts through convection with the gas temperature Ti at location i. At that location,

mass flow arrives from location j, whose temperature is Tj , and the mass flow leaves to

locations k and l, which are at temperature Tk and Tl respectively. The energy equation for

gases is (Equation (1.554)),

ρθ

∂2ψ

∂θ2

˙ θ + θ

∂2ψ

∂ρ−1∂θ

d : I −∇ ・ q + ρh = 0 (7.56)

where ψ(ρ

−1, θ). Similar to Equation (7.7), we expand ψ as a function of the temperature

T :

ψ(ρ

−1, θ) = ψ0(ρ

−1) − η0(ρ

−1)T − cv(ρ

−1, θ)

2θref

T 2 (7.57)

where

cv := ∂ε

∂θ

(7.58)

is the specific heat at constant volume for an ideal gas (Anderson 1989). Hence,

∂ψ

∂θ

= −η0 − cv

T

θref

+ O(T 2), T →0 (7.59)

∂2ψ

∂θ∂ρ−1

= − ∂η0

∂ρ−1

+ O(T ), T → 0 (7.60)

∂2ψ

∂θ2

= − cv

θref

+ O(T ), T →0. (7.61)

314 HEAT TRANSFER

Substituting into Equation (7.56) leads to (keeping only first-order terms)

ρcv

˙ T + θref

∂η0

∂ρ−1 d : I = −∇ ・ q + ρh. (7.62)

The function η0 can be further specified if we take the gas equation of state into account.

For an ideal gas, we have as the equation of state

p = Rρ(θref + T ) (7.63)

where R is the specific gas constant, and

p = − ∂ψ

∂ρ−1

= − ∂ψ0

∂ρ−1

+ ∂η0

∂ρ−1 T. (7.64)

Accordingly,

∂ψ0

∂ρ−1

= −Rρθref ⇒ ψ0 = Rθref ln ρ + C1 (7.65)

∂η0

∂ρ−1

= Rρ ⇒ η0 = −R ln ρ + C2. (7.66)

The heat equation, Equation (7.60), now yields

ρcv

˙ T + Rρθrefd : I = −∇ ・ q + ρh. (7.67)

Using Equation (1.517), this can also be written as

ρcv

T˙ = −∇ ・ q + ρh + Rθrefρ˙ (7.68)

or, since ρ = 1/v and θref ≈ θ,

ρcv

˙ T = −∇ ・ q + ρh − ρp ˙ v. (7.69)

Accordingly, a temperature increase can be obtained through heat influx or through mechanical

work (Anderson 1991). If we assume that the pressure p is constant, we have

Rρθ = constant (7.70)

or

Rρ˙θ + Rρ ˙ θ = 0. (7.71)

This can be transformed into

Rθref ˙ ρ ≈ Rθ ˙ρ = −Rρ ˙ θ = −Rρ ˙ T (7.72)

and

ρcv

˙ T − Rθref ˙ρ ≈ ρ ˙ T (cv + R) = ρ ˙ T cp (7.73)

HEAT TRANSFER 315

since the specific heat at constant pressure, cp, satisfies

R = cp − cv. (7.74)

Consequently, the energy equation for a gas reduces to

ρcp

˙ T = −qk

,k

+ ρh. (7.75)

The derivative of the temperature on the left-hand side is the total derivative consisting of

the local variation and the change due to convection:

˙ T = DT

Dt

= ∂T

∂t

+ T,kvk. (7.76)

So far, we dealt with solids, for which the convective term can be neglected. This is not

so for fluids and gases. Accordingly,

ρcp

 ∂T

∂t

+ T,kvk! = −qk

,k

+ ρh. (7.77)

The conservation of mass requires (Equation (1.223))

∂ρ

∂t

+ (ρvk),k = 0. (7.78)

Combining Equations (7.77) and (7.78) yields

cp

∂ρT

∂t

+ cp(Tρvk),k = −qk

,k

+ ρh. (7.79)

The gas nodes i, j, k, l, ・ ・ ・ stand for a given control volume that is fixed in space and

assigned to them. Integrating Equation (7.79) for node i yields

_

Vi

cp

∂ρT

∂t

dv + _

Vi

cp(Tρvk),k dv = −_

Vi

qk

,k dv + _

Vi

ρh dv. (7.80)

Transforming the volume integrals for the divergence terms to surface integrals (assuming

cp to be constant over the volume),

_

Vi

cp

∂ρT

∂t

dv + _

Ai

cpTρvk dak = −_

Ai

qk dak + _

Vi

ρh dv. (7.81)

We assume that the integrands of the volume integrals are constant across the volume:

_

Vi

cp

∂ρT

∂t

dv = cp(Ti)

∂ρ(Ti)Ti

∂t

Vi (7.82)

_

Vi

ρhdv = ρ(Ti)hiVi . (7.83)

The area of the convective term is split into areas with inflow and areas with outflow.

For both types, cp and T are assumed to be constant across the area. For inflow, T is the

316 HEAT TRANSFER

temperature of the neighboring node providing the flow; for outflow it is the temperature

of node i. Hence,

_

Ai

cpTρvk dak =_

j∈in

cp(Tj )Tj

_

Aij

ρvk dak + _

j∈out

cp(Ti)Ti

_

Aij

ρvk dak. (7.84)

The mass flow between node i and node j is defined by

˙m

ij = Ѓ}_

Aij

ρvk dak. (7.85)

The plus sign applies to the outflow and the minus sign to the inflow. Accordingly,

_

Ai

cpTρvk dak = −_

j∈in

cp(Tj )Tj ˙mij + _

j∈out

cp(Ti)Ti ˙mij . (7.86)

The first term on the right-hand side of Equation (7.81) relates to the convection from

the wall (surface Aiw) and the conduction in the fluid (surface Aif , Ai = Aif ∪ Aiw):

_

Ai

qk dak = _

Aiw

qk

conv dak + _

Aif

qk dak. (7.87)

The conduction in the fluid is neglected. Hence,

_

Ai

qk dak = −[h(Ti , T )(T − Ti)]Aiw. (7.88)

Summarizing, one obtains the following equation:

cp(Ti)

∂[ρ(Ti)Ti ]

∂t

Vi =_

j∈in

cp(Tj )Tj ˙mij − cp(Ti)Ti _

j∈out

˙m

ij

+ h(Ti , T )(T − Ti ) + mihi (7.89)

where

mi = ρ(Ti)Vi (7.90)

is the mass in the control volume and

h(Ti, T ) = h(Ti, T )Aiw (7.91)

A(Ti, T ) = A(Ti, T )Aiw. (7.92)

Equation (7.89) expresses that the change of heat energy at node i is caused by influx from

the other nodes, plus convection from the wall, minus outflux to the other nodes. In reality,

the inertia of the gas is small compared to the inertia of the wall. Consequently, the term

on the left-hand side of Equation (7.89) is usually neglected leading to

0 =_

j∈in

cp(Tj )Tj ˙mij − cp(Ti)Ti _

j∈out

˙m

ij + h(Ti , T )(T − Ti) + mihi . (7.93)

HEAT TRANSFER 317

This is a weakly nonlinear equation in the temperature:

h(Ti, T ) + cp(Ti) _

j∈out

˙m

ij

Ti −_

j∈in

_cp(Tj ) ˙mij _ Tj − h(Ti, T )T = mihi . (7.94)

Equation (7.94) can be considered as a nonlinear multiple-point constraint in the temperature,

analogous to the nonlinear displacement multiple-point constraints in Chapter 3. It

allows for the calculation of the gas temperatures as soon as the structural temperatures are

known.

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